<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="6">

<source>mol-hints.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13065 was started by mccune on cleo.thornwood,
Mon Jun 19 16:40:45 2006
The command was "/home/mccune/bin/prover9 -f mol-hints.in".
]]></heading>

<proof number="1" length="23" max_count="41">

<comments><![CDATA[
% Proof 1 at 0.20 (+ 0.00) seconds: absorb.
% Length of proof is 23.
% Level of proof is 13.
% Maximum clause weight is 0.
% Given clauses 35.
]]></comments>

  <clause id="494">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="497">
    <literal><![CDATA[
      f(f(c5,c5),f(c5,c6)) != c5
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="501">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(494(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="494 494"/>
    </justification>
  </clause>

  <clause id="502">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(494(a,1),501(a,1,1,2)),demod(494(12),494(12),494(14),494(17),494(30))].">
      <j1 rule="para" parents="494 501"/>
      <j2 rule="demod" parents="494 494 494 494 494"/>
    </justification>
  </clause>

  <clause id="503">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(494(a,1),502(a,1,2,2))].">
      <j1 rule="para" parents="494 502"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(494(a,1),503(a,1,2,1))].">
      <j1 rule="para" parents="494 503"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="507">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="509">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 501"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1)),demod(504(5),504(5))].">
      <j1 rule="para" parents="504 501"/>
      <j2 rule="demod" parents="504 504"/>
    </justification>
  </clause>

  <clause id="512">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(503(a,1),504(a,1,1)),demod(503(7)),flip(a)].">
      <j1 rule="para" parents="503 504"/>
      <j2 rule="demod" parents="503"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_demod(505),demod(510(7))].">
      <j1 rule="back_demod" parents="505"/>
      <j2 rule="demod" parents="510"/>
    </justification>
  </clause>

  <clause id="515">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(504(a,1),512(a,1,1,1)),demod(504(6))].">
      <j1 rule="para" parents="504 512"/>
      <j2 rule="demod" parents="504"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(515(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="515 494"/>
    </justification>
  </clause>

  <clause id="517">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(494(a,1),515(a,1,1))].">
      <j1 rule="para" parents="494 515"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(515(a,1),507(a,1,2,2,2))].">
      <j1 rule="para" parents="515 507"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(513(a,1),512(a,1)),flip(a)].">
      <j1 rule="para" parents="513 512"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(522(a,1),504(a,1,1)),demod(522(9)),flip(a)].">
      <j1 rule="para" parents="522 504"/>
      <j2 rule="demod" parents="522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="527">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(504(a,1),524(a,1,2,1)),demod(515(8))].">
      <j1 rule="para" parents="504 524"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(516(a,1),509(a,1,2,2,1)),demod(515(3),517(6),504(6),515(10),517(13),504(13),515(16),517(19),504(19),504(27),527(27),510(23),515(9),515(3),517(6),504(6)),flip(a)].">
      <j1 rule="para" parents="516 509"/>
      <j2 rule="demod" parents="515 517 504 515 517 504 515 517 504 504 527 510 515 515 517 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(504(a,1),535(a,1,1,1)),demod(515(6))].">
      <j1 rule="para" parents="504 535"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="540">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(524(a,1),521(a,1,1)),demod(536(16),515(13),510(11),515(3)),flip(a)].">
      <j1 rule="para" parents="524 521"/>
      <j2 rule="demod" parents="536 515 510 515"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="541">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(absorb)
    ]]></attribute>
    <justification jstring="[resolve(540,a,497,a)].">
      <j1 rule="resolve" parents="540 497"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="57" max_count="41">

<comments><![CDATA[
% Proof 2 at 0.42 (+ 0.00) seconds: one.
% Length of proof is 57.
% Level of proof is 22.
% Maximum clause weight is 0.
% Given clauses 63.
]]></comments>

  <clause id="494">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="496">
    <literal><![CDATA[
      f(c4,f(c4,c4)) != f(c3,f(c3,c3))
    ]]></literal>
    <attribute><![CDATA[
      answer(one)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="501">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(494(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="494 494"/>
    </justification>
  </clause>

  <clause id="502">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(494(a,1),501(a,1,1,2)),demod(494(12),494(12),494(14),494(17),494(30))].">
      <j1 rule="para" parents="494 501"/>
      <j2 rule="demod" parents="494 494 494 494 494"/>
    </justification>
  </clause>

  <clause id="503">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(494(a,1),502(a,1,2,2))].">
      <j1 rule="para" parents="494 502"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(494(a,1),503(a,1,2,1))].">
      <j1 rule="para" parents="494 503"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="506">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(504)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="507">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="509">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 501"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1)),demod(504(5),504(5))].">
      <j1 rule="para" parents="504 501"/>
      <j2 rule="demod" parents="504 504"/>
    </justification>
  </clause>

  <clause id="512">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(503(a,1),504(a,1,1)),demod(503(7)),flip(a)].">
      <j1 rule="para" parents="503 504"/>
      <j2 rule="demod" parents="503"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_demod(505),demod(510(7))].">
      <j1 rule="back_demod" parents="505"/>
      <j2 rule="demod" parents="510"/>
    </justification>
  </clause>

  <clause id="514">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(512(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="512 494"/>
    </justification>
  </clause>

  <clause id="515">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(504(a,1),512(a,1,1,1)),demod(504(6))].">
      <j1 rule="para" parents="504 512"/>
      <j2 rule="demod" parents="504"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(515(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="515 494"/>
    </justification>
  </clause>

  <clause id="517">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(494(a,1),515(a,1,1))].">
      <j1 rule="para" parents="494 515"/>
    </justification>
  </clause>

  <clause id="519">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(517)
    ]]></attribute>
    <justification jstring="[para(517(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="517 494"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(515(a,1),507(a,1,2,2,2))].">
      <j1 rule="para" parents="515 507"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(513(a,1),512(a,1)),flip(a)].">
      <j1 rule="para" parents="513 512"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="523">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(517(a,1),514(a,1,2,2,2))].">
      <j1 rule="para" parents="517 514"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(522(a,1),504(a,1,1)),demod(522(9)),flip(a)].">
      <j1 rule="para" parents="522 504"/>
      <j2 rule="demod" parents="522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="526">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(522)
    ]]></attribute>
    <justification jstring="[para(494(a,1),524(a,1,2,2))].">
      <j1 rule="para" parents="494 524"/>
    </justification>
  </clause>

  <clause id="527">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(504(a,1),524(a,1,2,1)),demod(515(8))].">
      <j1 rule="para" parents="504 524"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="528">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(524)
    ]]></attribute>
    <justification jstring="[para(515(a,1),524(a,1,2,2))].">
      <j1 rule="para" parents="515 524"/>
    </justification>
  </clause>

  <clause id="529">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(517(a,1),527(a,1,2,2))].">
      <j1 rule="para" parents="517 527"/>
    </justification>
  </clause>

  <clause id="530">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(504(a,1),523(a,1,1)),demod(515(3),515(5))].">
      <j1 rule="para" parents="504 523"/>
      <j2 rule="demod" parents="515 515"/>
    </justification>
  </clause>

  <clause id="531">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(530(a,1),517(a,1,2,1)),demod(530(18),515(11),530(18))].">
      <j1 rule="para" parents="530 517"/>
      <j2 rule="demod" parents="530 515 530"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(516(a,1),509(a,1,2,2,1)),demod(515(3),517(6),504(6),515(10),517(13),504(13),515(16),517(19),504(19),504(27),527(27),510(23),515(9),515(3),517(6),504(6)),flip(a)].">
      <j1 rule="para" parents="516 509"/>
      <j2 rule="demod" parents="515 517 504 515 517 504 515 517 504 504 527 510 515 515 517 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(504(a,1),535(a,1,1,1)),demod(515(6))].">
      <j1 rule="para" parents="504 535"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(536)
    ]]></attribute>
    <justification jstring="[para(517(a,1),535(a,1,2,2))].">
      <j1 rule="para" parents="517 535"/>
    </justification>
  </clause>

  <clause id="538">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(537)
    ]]></attribute>
    <justification jstring="[para(494(a,1),536(a,1,2,2))].">
      <j1 rule="para" parents="494 536"/>
    </justification>
  </clause>

  <clause id="539">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(538)
    ]]></attribute>
    <justification jstring="[para(515(a,1),536(a,1,2,2))].">
      <j1 rule="para" parents="515 536"/>
    </justification>
  </clause>

  <clause id="540">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(524(a,1),521(a,1,1)),demod(536(16),515(13),510(11),515(3)),flip(a)].">
      <j1 rule="para" parents="524 521"/>
      <j2 rule="demod" parents="536 515 510 515"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="543">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(542E)
    ]]></attribute>
    <justification jstring="[para(529(a,1),521(a,1,1)),demod(540(10),540(12),537(14),504(10)),flip(a)].">
      <j1 rule="para" parents="529 521"/>
      <j2 rule="demod" parents="540 540 537 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="544">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(543E)
    ]]></attribute>
    <justification jstring="[para(526(a,1),521(a,1,1)),demod(538(26),540(23),510(18),540(7)),flip(a)].">
      <j1 rule="para" parents="526 521"/>
      <j2 rule="demod" parents="538 540 510 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="545">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),x) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(544E)
    ]]></attribute>
    <justification jstring="[para(528(a,1),521(a,1,1)),demod(539(26),540(23),510(18),540(7)),flip(a)].">
      <j1 rule="para" parents="528 521"/>
      <j2 rule="demod" parents="539 540 510 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="547">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(548)
    ]]></attribute>
    <justification jstring="[para(517(a,1),540(a,1,2))].">
      <j1 rule="para" parents="517 540"/>
    </justification>
  </clause>

  <clause id="548">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,z)),f(y,z)),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(546)
    ]]></attribute>
    <justification jstring="[para(540(a,1),519(a,1,2,1))].">
      <j1 rule="para" parents="540 519"/>
    </justification>
  </clause>

  <clause id="550">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(550)
    ]]></attribute>
    <justification jstring="[back_demod(506),demod(547(3))].">
      <j1 rule="back_demod" parents="506"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="553">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(y,x),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(552)
    ]]></attribute>
    <justification jstring="[para(517(a,1),548(a,1,2))].">
      <j1 rule="para" parents="517 548"/>
    </justification>
  </clause>

  <clause id="554">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(y,x),z)),f(f(y,x),z)),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(553)
    ]]></attribute>
    <justification jstring="[para(504(a,1),553(a,1,2,1,1,1)),demod(547(3))].">
      <j1 rule="para" parents="504 553"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(557)
    ]]></attribute>
    <justification jstring="[para(494(a,1),554(a,1,2,1,1,2)),demod(494(12))].">
      <j1 rule="para" parents="494 554"/>
      <j2 rule="demod" parents="494"/>
    </justification>
  </clause>

  <clause id="559">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(x,x)))) = f(y,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(558)
    ]]></attribute>
    <justification jstring="[para(510(a,1),554(a,1,2,1,1,2)),demod(540(4),510(4))].">
      <j1 rule="para" parents="510 554"/>
      <j2 rule="demod" parents="540 510"/>
    </justification>
  </clause>

  <clause id="560">
    <literal><![CDATA[
      f(x,f(f(f(x,y),y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(559)
    ]]></attribute>
    <justification jstring="[para(515(a,1),554(a,1,2,1,1,2)),demod(515(4))].">
      <j1 rule="para" parents="515 554"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="561">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(560)
    ]]></attribute>
    <justification jstring="[para(545(a,1),558(a,1,2,2)),demod(545(9))].">
      <j1 rule="para" parents="545 558"/>
      <j2 rule="demod" parents="545"/>
    </justification>
  </clause>

  <clause id="562">
    <literal><![CDATA[
      f(f(x,x),x) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(561)
    ]]></attribute>
    <justification jstring="[para(512(a,1),560(a,1,2,1)),demod(540(5))].">
      <j1 rule="para" parents="512 560"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="563">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(562)
    ]]></attribute>
    <justification jstring="[back_demod(561),demod(562(2))].">
      <j1 rule="back_demod" parents="561"/>
      <j2 rule="demod" parents="562"/>
    </justification>
  </clause>

  <clause id="564">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),y))) = f(f(x,x),y)
    ]]></literal>
    <attribute><![CDATA[
      label(563)
    ]]></attribute>
    <justification jstring="[para(504(a,1),563(a,1,2,1,2)),demod(562(3))].">
      <j1 rule="para" parents="504 563"/>
      <j2 rule="demod" parents="562"/>
    </justification>
  </clause>

  <clause id="565">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(564)
    ]]></attribute>
    <justification jstring="[para(545(a,1),550(a,1,2,1))].">
      <j1 rule="para" parents="545 550"/>
    </justification>
  </clause>

  <clause id="566">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,f(x,x)),y)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(565)
    ]]></attribute>
    <justification jstring="[para(512(a,1),565(a,1,2,2,1,1)),demod(540(7))].">
      <j1 rule="para" parents="512 565"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="568">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,f(x,x)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(567D)
    ]]></attribute>
    <justification jstring="[para(531(a,1),566(a,1,2,2,2)),demod(515(17),531(14),544(7)),flip(a)].">
      <j1 rule="para" parents="531 566"/>
      <j2 rule="demod" parents="515 531 544"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="569">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(568)
    ]]></attribute>
    <justification jstring="[para(563(a,1),566(a,1,2,2))].">
      <j1 rule="para" parents="563 566"/>
    </justification>
  </clause>

  <clause id="571">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(570C)
    ]]></attribute>
    <justification jstring="[para(568(a,1),559(a,1,2,1)),demod(543(14),540(11)),flip(a)].">
      <j1 rule="para" parents="568 559"/>
      <j2 rule="demod" parents="543 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="578">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(576)
    ]]></attribute>
    <justification jstring="[para(571(a,1),564(a,1,2,2)),demod(571(14))].">
      <j1 rule="para" parents="571 564"/>
      <j2 rule="demod" parents="571"/>
    </justification>
  </clause>

  <clause id="579">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(577)
    ]]></attribute>
    <justification jstring="[para(571(a,1),569(a,1,2,2)),demod(578(7))].">
      <j1 rule="para" parents="571 569"/>
      <j2 rule="demod" parents="578"/>
    </justification>
  </clause>

  <clause id="580">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(one)
    ]]></attribute>
    <justification jstring="[resolve(579,a,496,a)].">
      <j1 rule="resolve" parents="579 496"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="53" max_count="41">

<comments><![CDATA[
% Proof 3 at 0.47 (+ 0.00) seconds: oml.
% Length of proof is 53.
% Level of proof is 21.
% Maximum clause weight is 0.
% Given clauses 68.
]]></comments>

  <clause id="494">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      f(c7,f(c7,f(c7,c8))) != f(c7,c8)
    ]]></literal>
    <attribute><![CDATA[
      answer(oml)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="501">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(494(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="494 494"/>
    </justification>
  </clause>

  <clause id="502">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(494(a,1),501(a,1,1,2)),demod(494(12),494(12),494(14),494(17),494(30))].">
      <j1 rule="para" parents="494 501"/>
      <j2 rule="demod" parents="494 494 494 494 494"/>
    </justification>
  </clause>

  <clause id="503">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(494(a,1),502(a,1,2,2))].">
      <j1 rule="para" parents="494 502"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(494(a,1),503(a,1,2,1))].">
      <j1 rule="para" parents="494 503"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="506">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(504)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="507">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="509">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 501"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1)),demod(504(5),504(5))].">
      <j1 rule="para" parents="504 501"/>
      <j2 rule="demod" parents="504 504"/>
    </justification>
  </clause>

  <clause id="512">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(503(a,1),504(a,1,1)),demod(503(7)),flip(a)].">
      <j1 rule="para" parents="503 504"/>
      <j2 rule="demod" parents="503"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_demod(505),demod(510(7))].">
      <j1 rule="back_demod" parents="505"/>
      <j2 rule="demod" parents="510"/>
    </justification>
  </clause>

  <clause id="514">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(512(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="512 494"/>
    </justification>
  </clause>

  <clause id="515">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(504(a,1),512(a,1,1,1)),demod(504(6))].">
      <j1 rule="para" parents="504 512"/>
      <j2 rule="demod" parents="504"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(515(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="515 494"/>
    </justification>
  </clause>

  <clause id="517">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(494(a,1),515(a,1,1))].">
      <j1 rule="para" parents="494 515"/>
    </justification>
  </clause>

  <clause id="519">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(517)
    ]]></attribute>
    <justification jstring="[para(517(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="517 494"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(515(a,1),507(a,1,2,2,2))].">
      <j1 rule="para" parents="515 507"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(513(a,1),512(a,1)),flip(a)].">
      <j1 rule="para" parents="513 512"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="523">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(517(a,1),514(a,1,2,2,2))].">
      <j1 rule="para" parents="517 514"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(522(a,1),504(a,1,1)),demod(522(9)),flip(a)].">
      <j1 rule="para" parents="522 504"/>
      <j2 rule="demod" parents="522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="526">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(522)
    ]]></attribute>
    <justification jstring="[para(494(a,1),524(a,1,2,2))].">
      <j1 rule="para" parents="494 524"/>
    </justification>
  </clause>

  <clause id="527">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(504(a,1),524(a,1,2,1)),demod(515(8))].">
      <j1 rule="para" parents="504 524"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="528">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(524)
    ]]></attribute>
    <justification jstring="[para(515(a,1),524(a,1,2,2))].">
      <j1 rule="para" parents="515 524"/>
    </justification>
  </clause>

  <clause id="529">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(517(a,1),527(a,1,2,2))].">
      <j1 rule="para" parents="517 527"/>
    </justification>
  </clause>

  <clause id="530">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(504(a,1),523(a,1,1)),demod(515(3),515(5))].">
      <j1 rule="para" parents="504 523"/>
      <j2 rule="demod" parents="515 515"/>
    </justification>
  </clause>

  <clause id="531">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(530(a,1),517(a,1,2,1)),demod(530(18),515(11),530(18))].">
      <j1 rule="para" parents="530 517"/>
      <j2 rule="demod" parents="530 515 530"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(516(a,1),509(a,1,2,2,1)),demod(515(3),517(6),504(6),515(10),517(13),504(13),515(16),517(19),504(19),504(27),527(27),510(23),515(9),515(3),517(6),504(6)),flip(a)].">
      <j1 rule="para" parents="516 509"/>
      <j2 rule="demod" parents="515 517 504 515 517 504 515 517 504 504 527 510 515 515 517 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(504(a,1),535(a,1,1,1)),demod(515(6))].">
      <j1 rule="para" parents="504 535"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(536)
    ]]></attribute>
    <justification jstring="[para(517(a,1),535(a,1,2,2))].">
      <j1 rule="para" parents="517 535"/>
    </justification>
  </clause>

  <clause id="538">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(537)
    ]]></attribute>
    <justification jstring="[para(494(a,1),536(a,1,2,2))].">
      <j1 rule="para" parents="494 536"/>
    </justification>
  </clause>

  <clause id="539">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(538)
    ]]></attribute>
    <justification jstring="[para(515(a,1),536(a,1,2,2))].">
      <j1 rule="para" parents="515 536"/>
    </justification>
  </clause>

  <clause id="540">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(524(a,1),521(a,1,1)),demod(536(16),515(13),510(11),515(3)),flip(a)].">
      <j1 rule="para" parents="524 521"/>
      <j2 rule="demod" parents="536 515 510 515"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="542">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(541E)
    ]]></attribute>
    <justification jstring="[para(527(a,1),521(a,1,1)),demod(540(10),540(12),535(14),504(10)),flip(a)].">
      <j1 rule="para" parents="527 521"/>
      <j2 rule="demod" parents="540 540 535 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="543">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(542E)
    ]]></attribute>
    <justification jstring="[para(529(a,1),521(a,1,1)),demod(540(10),540(12),537(14),504(10)),flip(a)].">
      <j1 rule="para" parents="529 521"/>
      <j2 rule="demod" parents="540 540 537 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="544">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(543E)
    ]]></attribute>
    <justification jstring="[para(526(a,1),521(a,1,1)),demod(538(26),540(23),510(18),540(7)),flip(a)].">
      <j1 rule="para" parents="526 521"/>
      <j2 rule="demod" parents="538 540 510 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="545">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),x) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(544E)
    ]]></attribute>
    <justification jstring="[para(528(a,1),521(a,1,1)),demod(539(26),540(23),510(18),540(7)),flip(a)].">
      <j1 rule="para" parents="528 521"/>
      <j2 rule="demod" parents="539 540 510 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="547">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(548)
    ]]></attribute>
    <justification jstring="[para(517(a,1),540(a,1,2))].">
      <j1 rule="para" parents="517 540"/>
    </justification>
  </clause>

  <clause id="548">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,z)),f(y,z)),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(546)
    ]]></attribute>
    <justification jstring="[para(540(a,1),519(a,1,2,1))].">
      <j1 rule="para" parents="540 519"/>
    </justification>
  </clause>

  <clause id="550">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(550)
    ]]></attribute>
    <justification jstring="[back_demod(506),demod(547(3))].">
      <j1 rule="back_demod" parents="506"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="553">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(y,x),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(552)
    ]]></attribute>
    <justification jstring="[para(517(a,1),548(a,1,2))].">
      <j1 rule="para" parents="517 548"/>
    </justification>
  </clause>

  <clause id="554">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(y,x),z)),f(f(y,x),z)),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(553)
    ]]></attribute>
    <justification jstring="[para(504(a,1),553(a,1,2,1,1,1)),demod(547(3))].">
      <j1 rule="para" parents="504 553"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="555">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(x,y),z)),f(f(x,y),z)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(554)
    ]]></attribute>
    <justification jstring="[para(515(a,1),553(a,1,2,1,1,1)),demod(540(3))].">
      <j1 rule="para" parents="515 553"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="559">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(x,x)))) = f(y,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(558)
    ]]></attribute>
    <justification jstring="[para(510(a,1),554(a,1,2,1,1,2)),demod(540(4),510(4))].">
      <j1 rule="para" parents="510 554"/>
      <j2 rule="demod" parents="540 510"/>
    </justification>
  </clause>

  <clause id="565">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(564)
    ]]></attribute>
    <justification jstring="[para(545(a,1),550(a,1,2,1))].">
      <j1 rule="para" parents="545 550"/>
    </justification>
  </clause>

  <clause id="566">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,f(x,x)),y)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(565)
    ]]></attribute>
    <justification jstring="[para(512(a,1),565(a,1,2,2,1,1)),demod(540(7))].">
      <j1 rule="para" parents="512 565"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="568">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,f(x,x)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(567D)
    ]]></attribute>
    <justification jstring="[para(531(a,1),566(a,1,2,2,2)),demod(515(17),531(14),544(7)),flip(a)].">
      <j1 rule="para" parents="531 566"/>
      <j2 rule="demod" parents="515 531 544"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="571">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(570C)
    ]]></attribute>
    <justification jstring="[para(568(a,1),559(a,1,2,1)),demod(543(14),540(11)),flip(a)].">
      <j1 rule="para" parents="568 559"/>
      <j2 rule="demod" parents="543 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="576">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(594)
    ]]></attribute>
    <justification jstring="[para(571(a,1),540(a,1,2))].">
      <j1 rule="para" parents="571 540"/>
    </justification>
  </clause>

  <clause id="577">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(575)
    ]]></attribute>
    <justification jstring="[para(571(a,1),542(a,1,2))].">
      <j1 rule="para" parents="571 542"/>
    </justification>
  </clause>

  <clause id="601">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(599)
    ]]></attribute>
    <justification jstring="[para(577(a,1),555(a,1,2,1,1)),demod(576(7))].">
      <j1 rule="para" parents="577 555"/>
      <j2 rule="demod" parents="576"/>
    </justification>
  </clause>

  <clause id="602">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(oml)
    ]]></attribute>
    <justification jstring="[resolve(601,a,498,a)].">
      <j1 rule="resolve" parents="601 498"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="89" max_count="41">

<comments><![CDATA[
% Proof 4 at 0.94 (+ 0.02) seconds: comm.
% Length of proof is 89.
% Level of proof is 34.
% Maximum clause weight is 0.
% Given clauses 119.
]]></comments>

  <clause id="494">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="495">
    <literal><![CDATA[
      f(c2,c1) != f(c1,c2)
    ]]></literal>
    <attribute><![CDATA[
      answer(comm)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="501">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(494(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="494 494"/>
    </justification>
  </clause>

  <clause id="502">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(494(a,1),501(a,1,1,2)),demod(494(12),494(12),494(14),494(17),494(30))].">
      <j1 rule="para" parents="494 501"/>
      <j2 rule="demod" parents="494 494 494 494 494"/>
    </justification>
  </clause>

  <clause id="503">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(494(a,1),502(a,1,2,2))].">
      <j1 rule="para" parents="494 502"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(494(a,1),503(a,1,2,1))].">
      <j1 rule="para" parents="494 503"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="506">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(504)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="507">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="508">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(506)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="504 501"/>
    </justification>
  </clause>

  <clause id="509">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 501"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1)),demod(504(5),504(5))].">
      <j1 rule="para" parents="504 501"/>
      <j2 rule="demod" parents="504 504"/>
    </justification>
  </clause>

  <clause id="512">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(503(a,1),504(a,1,1)),demod(503(7)),flip(a)].">
      <j1 rule="para" parents="503 504"/>
      <j2 rule="demod" parents="503"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_demod(505),demod(510(7))].">
      <j1 rule="back_demod" parents="505"/>
      <j2 rule="demod" parents="510"/>
    </justification>
  </clause>

  <clause id="514">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(512(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="512 494"/>
    </justification>
  </clause>

  <clause id="515">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(504(a,1),512(a,1,1,1)),demod(504(6))].">
      <j1 rule="para" parents="504 512"/>
      <j2 rule="demod" parents="504"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(515(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="515 494"/>
    </justification>
  </clause>

  <clause id="517">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(494(a,1),515(a,1,1))].">
      <j1 rule="para" parents="494 515"/>
    </justification>
  </clause>

  <clause id="519">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(517)
    ]]></attribute>
    <justification jstring="[para(517(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="517 494"/>
    </justification>
  </clause>

  <clause id="520">
    <literal><![CDATA[
      f(f(f(f(x,x),y),f(f(f(f(f(x,z),y),y),x),f(x,u))),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(518)
    ]]></attribute>
    <justification jstring="[para(494(a,1),517(a,1,2,1)),demod(494(20),494(22))].">
      <j1 rule="para" parents="494 517"/>
      <j2 rule="demod" parents="494 494"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(515(a,1),507(a,1,2,2,2))].">
      <j1 rule="para" parents="515 507"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(513(a,1),512(a,1)),flip(a)].">
      <j1 rule="para" parents="513 512"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="523">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(517(a,1),514(a,1,2,2,2))].">
      <j1 rule="para" parents="517 514"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(522(a,1),504(a,1,1)),demod(522(9)),flip(a)].">
      <j1 rule="para" parents="522 504"/>
      <j2 rule="demod" parents="522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="525">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),f(x,x)),f(f(x,f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(533)
    ]]></attribute>
    <justification jstring="[para(515(a,1),508(a,1,2,2,1,1))].">
      <j1 rule="para" parents="515 508"/>
    </justification>
  </clause>

  <clause id="526">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(522)
    ]]></attribute>
    <justification jstring="[para(494(a,1),524(a,1,2,2))].">
      <j1 rule="para" parents="494 524"/>
    </justification>
  </clause>

  <clause id="527">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(504(a,1),524(a,1,2,1)),demod(515(8))].">
      <j1 rule="para" parents="504 524"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="528">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(524)
    ]]></attribute>
    <justification jstring="[para(515(a,1),524(a,1,2,2))].">
      <j1 rule="para" parents="515 524"/>
    </justification>
  </clause>

  <clause id="529">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(517(a,1),527(a,1,2,2))].">
      <j1 rule="para" parents="517 527"/>
    </justification>
  </clause>

  <clause id="530">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(504(a,1),523(a,1,1)),demod(515(3),515(5))].">
      <j1 rule="para" parents="504 523"/>
      <j2 rule="demod" parents="515 515"/>
    </justification>
  </clause>

  <clause id="531">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(530(a,1),517(a,1,2,1)),demod(530(18),515(11),530(18))].">
      <j1 rule="para" parents="530 517"/>
      <j2 rule="demod" parents="530 515 530"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(516(a,1),509(a,1,2,2,1)),demod(515(3),517(6),504(6),515(10),517(13),504(13),515(16),517(19),504(19),504(27),527(27),510(23),515(9),515(3),517(6),504(6)),flip(a)].">
      <j1 rule="para" parents="516 509"/>
      <j2 rule="demod" parents="515 517 504 515 517 504 515 517 504 504 527 510 515 515 517 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(504(a,1),535(a,1,1,1)),demod(515(6))].">
      <j1 rule="para" parents="504 535"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(536)
    ]]></attribute>
    <justification jstring="[para(517(a,1),535(a,1,2,2))].">
      <j1 rule="para" parents="517 535"/>
    </justification>
  </clause>

  <clause id="538">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(537)
    ]]></attribute>
    <justification jstring="[para(494(a,1),536(a,1,2,2))].">
      <j1 rule="para" parents="494 536"/>
    </justification>
  </clause>

  <clause id="539">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(538)
    ]]></attribute>
    <justification jstring="[para(515(a,1),536(a,1,2,2))].">
      <j1 rule="para" parents="515 536"/>
    </justification>
  </clause>

  <clause id="540">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(524(a,1),521(a,1,1)),demod(536(16),515(13),510(11),515(3)),flip(a)].">
      <j1 rule="para" parents="524 521"/>
      <j2 rule="demod" parents="536 515 510 515"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="542">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(541E)
    ]]></attribute>
    <justification jstring="[para(527(a,1),521(a,1,1)),demod(540(10),540(12),535(14),504(10)),flip(a)].">
      <j1 rule="para" parents="527 521"/>
      <j2 rule="demod" parents="540 540 535 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="543">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(542E)
    ]]></attribute>
    <justification jstring="[para(529(a,1),521(a,1,1)),demod(540(10),540(12),537(14),504(10)),flip(a)].">
      <j1 rule="para" parents="529 521"/>
      <j2 rule="demod" parents="540 540 537 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="544">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(543E)
    ]]></attribute>
    <justification jstring="[para(526(a,1),521(a,1,1)),demod(538(26),540(23),510(18),540(7)),flip(a)].">
      <j1 rule="para" parents="526 521"/>
      <j2 rule="demod" parents="538 540 510 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="545">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),x) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(544E)
    ]]></attribute>
    <justification jstring="[para(528(a,1),521(a,1,1)),demod(539(26),540(23),510(18),540(7)),flip(a)].">
      <j1 rule="para" parents="528 521"/>
      <j2 rule="demod" parents="539 540 510 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="547">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(548)
    ]]></attribute>
    <justification jstring="[para(517(a,1),540(a,1,2))].">
      <j1 rule="para" parents="517 540"/>
    </justification>
  </clause>

  <clause id="548">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,z)),f(y,z)),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(546)
    ]]></attribute>
    <justification jstring="[para(540(a,1),519(a,1,2,1))].">
      <j1 rule="para" parents="540 519"/>
    </justification>
  </clause>

  <clause id="549">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),f(x,x)),f(f(x,f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(549)
    ]]></attribute>
    <justification jstring="[back_demod(525),demod(547(3))].">
      <j1 rule="back_demod" parents="525"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="550">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(550)
    ]]></attribute>
    <justification jstring="[back_demod(506),demod(547(3))].">
      <j1 rule="back_demod" parents="506"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="553">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(y,x),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(552)
    ]]></attribute>
    <justification jstring="[para(517(a,1),548(a,1,2))].">
      <j1 rule="para" parents="517 548"/>
    </justification>
  </clause>

  <clause id="554">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(y,x),z)),f(f(y,x),z)),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(553)
    ]]></attribute>
    <justification jstring="[para(504(a,1),553(a,1,2,1,1,1)),demod(547(3))].">
      <j1 rule="para" parents="504 553"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="555">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(x,y),z)),f(f(x,y),z)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(554)
    ]]></attribute>
    <justification jstring="[para(515(a,1),553(a,1,2,1,1,1)),demod(540(3))].">
      <j1 rule="para" parents="515 553"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(557)
    ]]></attribute>
    <justification jstring="[para(494(a,1),554(a,1,2,1,1,2)),demod(494(12))].">
      <j1 rule="para" parents="494 554"/>
      <j2 rule="demod" parents="494"/>
    </justification>
  </clause>

  <clause id="559">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(x,x)))) = f(y,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(558)
    ]]></attribute>
    <justification jstring="[para(510(a,1),554(a,1,2,1,1,2)),demod(540(4),510(4))].">
      <j1 rule="para" parents="510 554"/>
      <j2 rule="demod" parents="540 510"/>
    </justification>
  </clause>

  <clause id="560">
    <literal><![CDATA[
      f(x,f(f(f(x,y),y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(559)
    ]]></attribute>
    <justification jstring="[para(515(a,1),554(a,1,2,1,1,2)),demod(515(4))].">
      <j1 rule="para" parents="515 554"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="561">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(560)
    ]]></attribute>
    <justification jstring="[para(545(a,1),558(a,1,2,2)),demod(545(9))].">
      <j1 rule="para" parents="545 558"/>
      <j2 rule="demod" parents="545"/>
    </justification>
  </clause>

  <clause id="562">
    <literal><![CDATA[
      f(f(x,x),x) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(561)
    ]]></attribute>
    <justification jstring="[para(512(a,1),560(a,1,2,1)),demod(540(5))].">
      <j1 rule="para" parents="512 560"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="563">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(562)
    ]]></attribute>
    <justification jstring="[back_demod(561),demod(562(2))].">
      <j1 rule="back_demod" parents="561"/>
      <j2 rule="demod" parents="562"/>
    </justification>
  </clause>

  <clause id="564">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),y))) = f(f(x,x),y)
    ]]></literal>
    <attribute><![CDATA[
      label(563)
    ]]></attribute>
    <justification jstring="[para(504(a,1),563(a,1,2,1,2)),demod(562(3))].">
      <j1 rule="para" parents="504 563"/>
      <j2 rule="demod" parents="562"/>
    </justification>
  </clause>

  <clause id="565">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(564)
    ]]></attribute>
    <justification jstring="[para(545(a,1),550(a,1,2,1))].">
      <j1 rule="para" parents="545 550"/>
    </justification>
  </clause>

  <clause id="566">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,f(x,x)),y)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(565)
    ]]></attribute>
    <justification jstring="[para(512(a,1),565(a,1,2,2,1,1)),demod(540(7))].">
      <j1 rule="para" parents="512 565"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="568">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,f(x,x)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(567D)
    ]]></attribute>
    <justification jstring="[para(531(a,1),566(a,1,2,2,2)),demod(515(17),531(14),544(7)),flip(a)].">
      <j1 rule="para" parents="531 566"/>
      <j2 rule="demod" parents="515 531 544"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="569">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(568)
    ]]></attribute>
    <justification jstring="[para(563(a,1),566(a,1,2,2))].">
      <j1 rule="para" parents="563 566"/>
    </justification>
  </clause>

  <clause id="571">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(570C)
    ]]></attribute>
    <justification jstring="[para(568(a,1),559(a,1,2,1)),demod(543(14),540(11)),flip(a)].">
      <j1 rule="para" parents="568 559"/>
      <j2 rule="demod" parents="543 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="574">
    <literal><![CDATA[
      f(f(x,f(x,x)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(573)
    ]]></attribute>
    <justification jstring="[para(571(a,1),512(a,1,1))].">
      <j1 rule="para" parents="571 512"/>
    </justification>
  </clause>

  <clause id="576">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(594)
    ]]></attribute>
    <justification jstring="[para(571(a,1),540(a,1,2))].">
      <j1 rule="para" parents="571 540"/>
    </justification>
  </clause>

  <clause id="577">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(575)
    ]]></attribute>
    <justification jstring="[para(571(a,1),542(a,1,2))].">
      <j1 rule="para" parents="571 542"/>
    </justification>
  </clause>

  <clause id="578">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(576)
    ]]></attribute>
    <justification jstring="[para(571(a,1),564(a,1,2,2)),demod(571(14))].">
      <j1 rule="para" parents="571 564"/>
      <j2 rule="demod" parents="571"/>
    </justification>
  </clause>

  <clause id="579">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(577)
    ]]></attribute>
    <justification jstring="[para(571(a,1),569(a,1,2,2)),demod(578(7))].">
      <j1 rule="para" parents="571 569"/>
      <j2 rule="demod" parents="578"/>
    </justification>
  </clause>

  <clause id="581">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(z,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(579)
    ]]></attribute>
    <justification jstring="[para(574(a,2),504(a,1,2))].">
      <j1 rule="para" parents="574 504"/>
    </justification>
  </clause>

  <clause id="583">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(z,x))) = f(z,x)
    ]]></literal>
    <attribute><![CDATA[
      label(581)
    ]]></attribute>
    <justification jstring="[para(574(a,2),517(a,1,2))].">
      <j1 rule="para" parents="574 517"/>
    </justification>
  </clause>

  <clause id="590">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),y),f(y,z)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583)
    ]]></attribute>
    <justification jstring="[para(574(a,2),540(a,1,1))].">
      <j1 rule="para" parents="574 540"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(590)
    ]]></attribute>
    <justification jstring="[para(574(a,1),569(a,1,2))].">
      <j1 rule="para" parents="574 569"/>
    </justification>
  </clause>

  <clause id="599">
    <literal><![CDATA[
      f(f(x,y),f(f(y,x),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(597)
    ]]></attribute>
    <justification jstring="[para(577(a,1),553(a,1,2,1,1)),demod(576(7))].">
      <j1 rule="para" parents="577 553"/>
      <j2 rule="demod" parents="576"/>
    </justification>
  </clause>

  <clause id="600">
    <literal><![CDATA[
      f(x,f(x,f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(598)
    ]]></attribute>
    <justification jstring="[para(577(a,1),554(a,1,2,1,1)),demod(576(7))].">
      <j1 rule="para" parents="577 554"/>
      <j2 rule="demod" parents="576"/>
    </justification>
  </clause>

  <clause id="601">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(599)
    ]]></attribute>
    <justification jstring="[para(577(a,1),555(a,1,2,1,1)),demod(576(7))].">
      <j1 rule="para" parents="577 555"/>
      <j2 rule="demod" parents="576"/>
    </justification>
  </clause>

  <clause id="610">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(f(f(f(y,z),y),y),y),f(y,u))),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(607)
    ]]></attribute>
    <justification jstring="[para(579(a,1),520(a,1,1,1)),demod(540(6),540(7))].">
      <j1 rule="para" parents="579 520"/>
      <j2 rule="demod" parents="540 540"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,f(x,y)),x),f(x,f(x,y)))) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(616)
    ]]></attribute>
    <justification jstring="[para(601(a,1),599(a,1,1))].">
      <j1 rule="para" parents="601 599"/>
    </justification>
  </clause>

  <clause id="628">
    <literal><![CDATA[
      f(f(f(x,y),x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(620)
    ]]></attribute>
    <justification jstring="[para(590(a,1),549(a,1,2,1,1,1)),demod(590(6),593(6),590(9),583(9),590(9))].">
      <j1 rule="para" parents="590 549"/>
      <j2 rule="demod" parents="590 593 590 583 590"/>
    </justification>
  </clause>

  <clause id="629">
    <literal><![CDATA[
      f(f(x,y),x) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(623)
    ]]></attribute>
    <justification jstring="[back_demod(624),demod(628(7))].">
      <j1 rule="back_demod" parents="624"/>
      <j2 rule="demod" parents="628"/>
    </justification>
  </clause>

  <clause id="632">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,f(y,z)),f(y,u))),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(626)
    ]]></attribute>
    <justification jstring="[back_demod(610),demod(629(4),629(5),601(5),629(4))].">
      <j1 rule="back_demod" parents="610"/>
      <j2 rule="demod" parents="629 629 601 629"/>
    </justification>
  </clause>

  <clause id="655">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,z),f(z,u))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(644)
    ]]></attribute>
    <justification jstring="[para(600(a,1),632(a,1,1,2,1))].">
      <j1 rule="para" parents="600 632"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,z),f(u,z))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(646)
    ]]></attribute>
    <justification jstring="[para(517(a,1),655(a,1,1,2,2))].">
      <j1 rule="para" parents="517 655"/>
    </justification>
  </clause>

  <clause id="660">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),y) = f(f(u,f(u,u)),f(f(x,y),f(z,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(649)
    ]]></attribute>
    <justification jstring="[para(657(a,1),540(a,1,2)),demod(581(13))].">
      <j1 rule="para" parents="657 540"/>
      <j2 rule="demod" parents="581"/>
    </justification>
  </clause>

  <clause id="670">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),y) = f(y,f(f(x,y),f(z,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(664B)
    ]]></attribute>
    <justification jstring="[para(660(a,1),517(a,1,2,1)),demod(590(11)),flip(a)].">
      <j1 rule="para" parents="660 517"/>
      <j2 rule="demod" parents="590"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="672">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,y),f(x,y)))) = f(f(z,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(666)
    ]]></attribute>
    <justification jstring="[para(660(a,2),583(a,1,2)),demod(670(5))].">
      <j1 rule="para" parents="660 583"/>
      <j2 rule="demod" parents="670"/>
    </justification>
  </clause>

  <clause id="673">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x,y)),x))) = f(f(z,f(x,y)),x)
    ]]></literal>
    <attribute><![CDATA[
      label(669)
    ]]></attribute>
    <justification jstring="[para(540(a,1),672(a,1,1)),demod(540(6),540(11))].">
      <j1 rule="para" parents="540 672"/>
      <j2 rule="demod" parents="540 540"/>
    </justification>
  </clause>

  <clause id="675">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,z),f(x,y)))) = f(f(y,z),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(668)
    ]]></attribute>
    <justification jstring="[para(545(a,1),672(a,1,2,2,1)),demod(545(10))].">
      <j1 rule="para" parents="545 672"/>
      <j2 rule="demod" parents="545"/>
    </justification>
  </clause>

  <clause id="676">
    <literal><![CDATA[
      f(x,f(f(x,y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(670)
    ]]></attribute>
    <justification jstring="[para(547(a,1),673(a,1,2,2,1)),demod(547(7))].">
      <j1 rule="para" parents="547 673"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="677">
    <literal><![CDATA[
      f(f(x,y),f(y,x)) = f(f(y,x),f(y,x))
    ]]></literal>
    <attribute><![CDATA[
      label(671A)
    ]]></attribute>
    <justification jstring="[para(676(a,1),675(a,1,2)),flip(a)].">
      <j1 rule="para" parents="676 675"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="678">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(z,y),f(y,z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(672)
    ]]></attribute>
    <justification jstring="[para(677(a,2),504(a,1,2))].">
      <j1 rule="para" parents="677 504"/>
    </justification>
  </clause>

  <clause id="680">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(933C)
    ]]></attribute>
    <justification jstring="[para(677(a,1),677(a,1,1)),demod(540(7),678(8))].">
      <j1 rule="para" parents="677 677"/>
      <j2 rule="demod" parents="540 678"/>
    </justification>
  </clause>

  <clause id="681">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(comm)
    ]]></attribute>
    <justification jstring="[resolve(680,a,495,a)].">
      <j1 rule="resolve" parents="680 495"/>
    </justification>
  </clause>

</proof>

<proof number="5" length="164" max_count="41">

<comments><![CDATA[
% Proof 5 at 4.82 (+ 0.05) seconds: assoc.
% Length of proof is 164.
% Level of proof is 59.
% Maximum clause weight is 0.
% Given clauses 237.
]]></comments>

  <clause id="6">
    <literal><![CDATA[
      f(c11,f(f(c10,c9),f(c10,c9))) != f(c9,f(f(c10,c11),f(c10,c11)))
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="494">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(492)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="499">
    <literal><![CDATA[
      f(c9,f(f(c10,c11),f(c10,c11))) != f(c11,f(f(c10,c9),f(c10,c9)))
    ]]></literal>
    <attribute><![CDATA[
      answer(assoc)
    ]]></attribute>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="501">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),u),f(f(f(f(f(f(y,z),x),u),u),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(499)
    ]]></attribute>
    <justification jstring="[para(494(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="494 494"/>
    </justification>
  </clause>

  <clause id="502">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(f(f(y,y),u),f(f(f(f(f(y,v),u),u),y),f(y,w)))))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(500)
    ]]></attribute>
    <justification jstring="[para(494(a,1),501(a,1,1,2)),demod(494(12),494(12),494(14),494(17),494(30))].">
      <j1 rule="para" parents="494 501"/>
      <j2 rule="demod" parents="494 494 494 494 494"/>
    </justification>
  </clause>

  <clause id="503">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508A)
    ]]></attribute>
    <justification jstring="[para(494(a,1),502(a,1,2,2))].">
      <j1 rule="para" parents="494 502"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      f(f(x,y),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(508B)
    ]]></attribute>
    <justification jstring="[para(494(a,1),503(a,1,2,1))].">
      <j1 rule="para" parents="494 503"/>
    </justification>
  </clause>

  <clause id="505">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,y)),f(y,y)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(503)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="506">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(504)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="507">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(x,x)),f(f(f(x,f(x,x)),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),494(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 494"/>
    </justification>
  </clause>

  <clause id="508">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(506)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1,1,1,1))].">
      <j1 rule="para" parents="504 501"/>
    </justification>
  </clause>

  <clause id="509">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586A)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1,1,1))].">
      <j1 rule="para" parents="504 501"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      f(f(x,f(y,y)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(508)
    ]]></attribute>
    <justification jstring="[para(504(a,1),501(a,1,2,2,1)),demod(504(5),504(5))].">
      <j1 rule="para" parents="504 501"/>
      <j2 rule="demod" parents="504 504"/>
    </justification>
  </clause>

  <clause id="512">
    <literal><![CDATA[
      f(f(f(x,x),y),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(510B)
    ]]></attribute>
    <justification jstring="[para(503(a,1),504(a,1,1)),demod(503(7)),flip(a)].">
      <j1 rule="para" parents="503 504"/>
      <j2 rule="demod" parents="503"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="513">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(511)
    ]]></attribute>
    <justification jstring="[back_demod(505),demod(510(7))].">
      <j1 rule="back_demod" parents="505"/>
      <j2 rule="demod" parents="510"/>
    </justification>
  </clause>

  <clause id="514">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(512)
    ]]></attribute>
    <justification jstring="[para(512(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="512 494"/>
    </justification>
  </clause>

  <clause id="515">
    <literal><![CDATA[
      f(f(x,y),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(513)
    ]]></attribute>
    <justification jstring="[para(504(a,1),512(a,1,1,1)),demod(504(6))].">
      <j1 rule="para" parents="504 512"/>
      <j2 rule="demod" parents="504"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),f(f(y,x),f(y,x))),f(f(f(y,x),y),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(514)
    ]]></attribute>
    <justification jstring="[para(515(a,1),494(a,1,2,2,1,1))].">
      <j1 rule="para" parents="515 494"/>
    </justification>
  </clause>

  <clause id="517">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(515)
    ]]></attribute>
    <justification jstring="[para(494(a,1),515(a,1,1))].">
      <j1 rule="para" parents="494 515"/>
    </justification>
  </clause>

  <clause id="518">
    <literal><![CDATA[
      f(x,f(f(x,y),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(516)
    ]]></attribute>
    <justification jstring="[para(515(a,1),515(a,1,1))].">
      <j1 rule="para" parents="515 515"/>
    </justification>
  </clause>

  <clause id="519">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(y,x),z),z),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(517)
    ]]></attribute>
    <justification jstring="[para(517(a,1),494(a,1,2,2,2))].">
      <j1 rule="para" parents="517 494"/>
    </justification>
  </clause>

  <clause id="520">
    <literal><![CDATA[
      f(f(f(f(x,x),y),f(f(f(f(f(x,z),y),y),x),f(x,u))),f(x,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(518)
    ]]></attribute>
    <justification jstring="[para(494(a,1),517(a,1,2,1)),demod(494(20),494(22))].">
      <j1 rule="para" parents="494 517"/>
      <j2 rule="demod" parents="494 494"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(f(f(y,z),f(y,z)),f(x,x)),f(f(f(x,f(x,x)),f(y,z)),y))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(528)
    ]]></attribute>
    <justification jstring="[para(515(a,1),507(a,1,2,2,2))].">
      <j1 rule="para" parents="515 507"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      f(f(x,f(f(x,x),f(x,y))),f(x,f(f(x,x),f(x,y)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(519A)
    ]]></attribute>
    <justification jstring="[para(513(a,1),512(a,1)),flip(a)].">
      <j1 rule="para" parents="513 512"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="523">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),x),f(f(f(x,x),x),f(y,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(520)
    ]]></attribute>
    <justification jstring="[para(517(a,1),514(a,1,2,2,2))].">
      <j1 rule="para" parents="517 514"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      f(x,f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(521B)
    ]]></attribute>
    <justification jstring="[para(522(a,1),504(a,1,1)),demod(522(9)),flip(a)].">
      <j1 rule="para" parents="522 504"/>
      <j2 rule="demod" parents="522"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="525">
    <literal><![CDATA[
      f(f(f(x,x),f(y,x)),f(f(f(f(y,x),f(y,x)),f(x,x)),f(f(x,f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(533)
    ]]></attribute>
    <justification jstring="[para(515(a,1),508(a,1,2,2,1,1))].">
      <j1 rule="para" parents="515 508"/>
    </justification>
  </clause>

  <clause id="526">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(522)
    ]]></attribute>
    <justification jstring="[para(494(a,1),524(a,1,2,2))].">
      <j1 rule="para" parents="494 524"/>
    </justification>
  </clause>

  <clause id="527">
    <literal><![CDATA[
      f(f(x,x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(523)
    ]]></attribute>
    <justification jstring="[para(504(a,1),524(a,1,2,1)),demod(515(8))].">
      <j1 rule="para" parents="504 524"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="528">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(524)
    ]]></attribute>
    <justification jstring="[para(515(a,1),524(a,1,2,2))].">
      <j1 rule="para" parents="515 524"/>
    </justification>
  </clause>

  <clause id="529">
    <literal><![CDATA[
      f(f(x,x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(517(a,1),527(a,1,2,2))].">
      <j1 rule="para" parents="517 527"/>
    </justification>
  </clause>

  <clause id="530">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x))))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(526)
    ]]></attribute>
    <justification jstring="[para(504(a,1),523(a,1,1)),demod(515(3),515(5))].">
      <j1 rule="para" parents="504 523"/>
      <j2 rule="demod" parents="515 515"/>
    </justification>
  </clause>

  <clause id="531">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(x,f(x,x)),f(y,f(x,x)))),x) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(527)
    ]]></attribute>
    <justification jstring="[para(530(a,1),517(a,1,2,1)),demod(530(18),515(11),530(18))].">
      <j1 rule="para" parents="530 517"/>
      <j2 rule="demod" parents="530 515 530"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(f(x,x),y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(535E)
    ]]></attribute>
    <justification jstring="[para(516(a,1),509(a,1,2,2,1)),demod(515(3),517(6),504(6),515(10),517(13),504(13),515(16),517(19),504(19),504(27),527(27),510(23),515(9),515(3),517(6),504(6)),flip(a)].">
      <j1 rule="para" parents="516 509"/>
      <j2 rule="demod" parents="515 517 504 515 517 504 515 517 504 504 527 510 515 515 517 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,x),f(x,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(534F)
    ]]></attribute>
    <justification jstring="[para(504(a,1),535(a,1,1,1)),demod(515(6))].">
      <j1 rule="para" parents="504 535"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      f(f(f(x,x),x),f(x,f(y,f(x,x)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(536)
    ]]></attribute>
    <justification jstring="[para(517(a,1),535(a,1,2,2))].">
      <j1 rule="para" parents="517 535"/>
    </justification>
  </clause>

  <clause id="538">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),y)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(537)
    ]]></attribute>
    <justification jstring="[para(494(a,1),536(a,1,2,2))].">
      <j1 rule="para" parents="494 536"/>
    </justification>
  </clause>

  <clause id="539">
    <literal><![CDATA[
      f(f(f(x,y),f(f(x,y),f(x,y))),f(f(f(x,y),f(x,y)),x)) = f(f(x,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(538)
    ]]></attribute>
    <justification jstring="[para(515(a,1),536(a,1,2,2))].">
      <j1 rule="para" parents="515 536"/>
    </justification>
  </clause>

  <clause id="540">
    <literal><![CDATA[
      f(f(x,x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(525)
    ]]></attribute>
    <justification jstring="[para(524(a,1),521(a,1,1)),demod(536(16),515(13),510(11),515(3)),flip(a)].">
      <j1 rule="para" parents="524 521"/>
      <j2 rule="demod" parents="536 515 510 515"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="542">
    <literal><![CDATA[
      f(x,f(f(x,x),y)) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(541E)
    ]]></attribute>
    <justification jstring="[para(527(a,1),521(a,1,1)),demod(540(10),540(12),535(14),504(10)),flip(a)].">
      <j1 rule="para" parents="527 521"/>
      <j2 rule="demod" parents="540 540 535 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="543">
    <literal><![CDATA[
      f(x,f(y,f(x,x))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(542E)
    ]]></attribute>
    <justification jstring="[para(529(a,1),521(a,1,1)),demod(540(10),540(12),537(14),504(10)),flip(a)].">
      <j1 rule="para" parents="529 521"/>
      <j2 rule="demod" parents="540 540 537 504"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="544">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),y) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(543E)
    ]]></attribute>
    <justification jstring="[para(526(a,1),521(a,1,1)),demod(538(26),540(23),510(18),540(7)),flip(a)].">
      <j1 rule="para" parents="526 521"/>
      <j2 rule="demod" parents="538 540 510 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="545">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),x) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(544E)
    ]]></attribute>
    <justification jstring="[para(528(a,1),521(a,1,1)),demod(539(26),540(23),510(18),540(7)),flip(a)].">
      <j1 rule="para" parents="528 521"/>
      <j2 rule="demod" parents="539 540 510 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="547">
    <literal><![CDATA[
      f(f(x,x),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(548)
    ]]></attribute>
    <justification jstring="[para(517(a,1),540(a,1,2))].">
      <j1 rule="para" parents="517 540"/>
    </justification>
  </clause>

  <clause id="548">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(f(f(f(y,x),f(y,z)),f(y,z)),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(546)
    ]]></attribute>
    <justification jstring="[para(540(a,1),519(a,1,2,1))].">
      <j1 rule="para" parents="540 519"/>
    </justification>
  </clause>

  <clause id="549">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),f(x,x)),f(f(x,f(y,x)),x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(549)
    ]]></attribute>
    <justification jstring="[back_demod(525),demod(547(3))].">
      <j1 rule="back_demod" parents="525"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="550">
    <literal><![CDATA[
      f(x,f(f(f(f(y,x),f(y,x)),z),f(f(f(f(x,z),z),f(y,x)),f(f(y,x),u)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(550)
    ]]></attribute>
    <justification jstring="[back_demod(506),demod(547(3))].">
      <j1 rule="back_demod" parents="506"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="552">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(f(f(f(x,y),z),z),y),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(551)
    ]]></attribute>
    <justification jstring="[para(544(a,1),494(a,1,1)),demod(517(7))].">
      <j1 rule="para" parents="544 494"/>
      <j2 rule="demod" parents="517"/>
    </justification>
  </clause>

  <clause id="553">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(y,x),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(552)
    ]]></attribute>
    <justification jstring="[para(517(a,1),548(a,1,2))].">
      <j1 rule="para" parents="517 548"/>
    </justification>
  </clause>

  <clause id="554">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(y,x),z)),f(f(y,x),z)),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(553)
    ]]></attribute>
    <justification jstring="[para(504(a,1),553(a,1,2,1,1,1)),demod(547(3))].">
      <j1 rule="para" parents="504 553"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="555">
    <literal><![CDATA[
      f(x,f(f(f(x,f(f(x,y),z)),f(f(x,y),z)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(554)
    ]]></attribute>
    <justification jstring="[para(515(a,1),553(a,1,2,1,1,1)),demod(540(3))].">
      <j1 rule="para" parents="515 553"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="556">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(x,y),f(y,z)),f(y,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(555)
    ]]></attribute>
    <justification jstring="[para(517(a,1),553(a,1,2,1,1,1)),demod(544(4))].">
      <j1 rule="para" parents="517 553"/>
      <j2 rule="demod" parents="544"/>
    </justification>
  </clause>

  <clause id="557">
    <literal><![CDATA[
      f(f(x,y),f(f(f(f(x,y),f(z,y)),f(z,y)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(556)
    ]]></attribute>
    <justification jstring="[para(517(a,1),556(a,1,2,1,1,2)),demod(517(8))].">
      <j1 rule="para" parents="517 556"/>
      <j2 rule="demod" parents="517"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(557)
    ]]></attribute>
    <justification jstring="[para(494(a,1),554(a,1,2,1,1,2)),demod(494(12))].">
      <j1 rule="para" parents="494 554"/>
      <j2 rule="demod" parents="494"/>
    </justification>
  </clause>

  <clause id="559">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(x,x)))) = f(y,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(558)
    ]]></attribute>
    <justification jstring="[para(510(a,1),554(a,1,2,1,1,2)),demod(540(4),510(4))].">
      <j1 rule="para" parents="510 554"/>
      <j2 rule="demod" parents="540 510"/>
    </justification>
  </clause>

  <clause id="560">
    <literal><![CDATA[
      f(x,f(f(f(x,y),y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(559)
    ]]></attribute>
    <justification jstring="[para(515(a,1),554(a,1,2,1,1,2)),demod(515(4))].">
      <j1 rule="para" parents="515 554"/>
      <j2 rule="demod" parents="515"/>
    </justification>
  </clause>

  <clause id="561">
    <literal><![CDATA[
      f(x,f(f(f(x,x),x),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(560)
    ]]></attribute>
    <justification jstring="[para(545(a,1),558(a,1,2,2)),demod(545(9))].">
      <j1 rule="para" parents="545 558"/>
      <j2 rule="demod" parents="545"/>
    </justification>
  </clause>

  <clause id="562">
    <literal><![CDATA[
      f(f(x,x),x) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(561)
    ]]></attribute>
    <justification jstring="[para(512(a,1),560(a,1,2,1)),demod(540(5))].">
      <j1 rule="para" parents="512 560"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="563">
    <literal><![CDATA[
      f(x,f(f(x,f(x,x)),f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(562)
    ]]></attribute>
    <justification jstring="[back_demod(561),demod(562(2))].">
      <j1 rule="back_demod" parents="561"/>
      <j2 rule="demod" parents="562"/>
    </justification>
  </clause>

  <clause id="564">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(f(x,x),y))) = f(f(x,x),y)
    ]]></literal>
    <attribute><![CDATA[
      label(563)
    ]]></attribute>
    <justification jstring="[para(504(a,1),563(a,1,2,1,2)),demod(562(3))].">
      <j1 rule="para" parents="504 563"/>
      <j2 rule="demod" parents="562"/>
    </justification>
  </clause>

  <clause id="565">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(564)
    ]]></attribute>
    <justification jstring="[para(545(a,1),550(a,1,2,1))].">
      <j1 rule="para" parents="545 550"/>
    </justification>
  </clause>

  <clause id="566">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,f(f(x,f(x,x)),y)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(565)
    ]]></attribute>
    <justification jstring="[para(512(a,1),565(a,1,2,2,1,1)),demod(540(7))].">
      <j1 rule="para" parents="512 565"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="567">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(f(x,y),y),f(y,x)),f(z,f(y,x))))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(566)
    ]]></attribute>
    <justification jstring="[para(517(a,1),565(a,1,2,2,2))].">
      <j1 rule="para" parents="517 565"/>
    </justification>
  </clause>

  <clause id="568">
    <literal><![CDATA[
      f(f(x,f(x,x)),f(f(x,f(x,x)),f(x,f(x,x)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(567D)
    ]]></attribute>
    <justification jstring="[para(531(a,1),566(a,1,2,2,2)),demod(515(17),531(14),544(7)),flip(a)].">
      <j1 rule="para" parents="531 566"/>
      <j2 rule="demod" parents="515 531 544"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="569">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(568)
    ]]></attribute>
    <justification jstring="[para(563(a,1),566(a,1,2,2))].">
      <j1 rule="para" parents="563 566"/>
    </justification>
  </clause>

  <clause id="570">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),f(f(f(x,y),f(f(x,y),f(x,y))),x)) = f(f(x,y),f(f(x,y),f(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(567B)
    ]]></attribute>
    <justification jstring="[para(515(a,1),569(a,1,2,2))].">
      <j1 rule="para" parents="515 569"/>
    </justification>
  </clause>

  <clause id="571">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(570C)
    ]]></attribute>
    <justification jstring="[para(568(a,1),559(a,1,2,1)),demod(543(14),540(11)),flip(a)].">
      <j1 rule="para" parents="568 559"/>
      <j2 rule="demod" parents="543 540"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="574">
    <literal><![CDATA[
      f(f(x,f(x,x)),y) = f(y,y)
    ]]></literal>
    <attribute><![CDATA[
      label(573)
    ]]></attribute>
    <justification jstring="[para(571(a,1),512(a,1,1))].">
      <j1 rule="para" parents="571 512"/>
    </justification>
  </clause>

  <clause id="576">
    <literal><![CDATA[
      f(f(x,x),f(y,f(y,y))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(594)
    ]]></attribute>
    <justification jstring="[para(571(a,1),540(a,1,2))].">
      <j1 rule="para" parents="571 540"/>
    </justification>
  </clause>

  <clause id="577">
    <literal><![CDATA[
      f(x,f(y,f(y,y))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(575)
    ]]></attribute>
    <justification jstring="[para(571(a,1),542(a,1,2))].">
      <j1 rule="para" parents="571 542"/>
    </justification>
  </clause>

  <clause id="578">
    <literal><![CDATA[
      f(f(x,x),f(f(x,f(x,x)),f(y,f(y,y)))) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(576)
    ]]></attribute>
    <justification jstring="[para(571(a,1),564(a,1,2,2)),demod(571(14))].">
      <j1 rule="para" parents="571 564"/>
      <j2 rule="demod" parents="571"/>
    </justification>
  </clause>

  <clause id="579">
    <literal><![CDATA[
      f(x,f(x,x)) = f(y,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(577)
    ]]></attribute>
    <justification jstring="[para(571(a,1),569(a,1,2,2)),demod(578(7))].">
      <j1 rule="para" parents="571 569"/>
      <j2 rule="demod" parents="578"/>
    </justification>
  </clause>

  <clause id="581">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(z,z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(579)
    ]]></attribute>
    <justification jstring="[para(574(a,2),504(a,1,2))].">
      <j1 rule="para" parents="574 504"/>
    </justification>
  </clause>

  <clause id="583">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(z,x))) = f(z,x)
    ]]></literal>
    <attribute><![CDATA[
      label(581)
    ]]></attribute>
    <justification jstring="[para(574(a,2),517(a,1,2))].">
      <j1 rule="para" parents="574 517"/>
    </justification>
  </clause>

  <clause id="586">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(x,z))) = f(x,z)
    ]]></literal>
    <attribute><![CDATA[
      label(582)
    ]]></attribute>
    <justification jstring="[para(574(a,2),518(a,1,2))].">
      <j1 rule="para" parents="574 518"/>
    </justification>
  </clause>

  <clause id="587">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(y,z)),z) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(586)
    ]]></attribute>
    <justification jstring="[para(574(a,1),509(a,1,2,1,2)),demod(571(13),571(14),586(13))].">
      <j1 rule="para" parents="574 509"/>
      <j2 rule="demod" parents="571 571 586"/>
    </justification>
  </clause>

  <clause id="589">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(u,f(u,u)),f(f(f(f(y,x),z),z),y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(585)
    ]]></attribute>
    <justification jstring="[para(574(a,2),519(a,1,2,2))].">
      <j1 rule="para" parents="574 519"/>
    </justification>
  </clause>

  <clause id="590">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),y),f(y,z)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(583)
    ]]></attribute>
    <justification jstring="[para(574(a,2),540(a,1,1))].">
      <j1 rule="para" parents="574 540"/>
    </justification>
  </clause>

  <clause id="592">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),z)) = f(x,f(z,z))
    ]]></literal>
    <attribute><![CDATA[
      label(589B)
    ]]></attribute>
    <justification jstring="[para(574(a,2),559(a,2,2)),demod(583(7)),flip(a)].">
      <j1 rule="para" parents="574 559"/>
      <j2 rule="demod" parents="583"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      f(f(x,x),f(f(x,y),f(x,y))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(590)
    ]]></attribute>
    <justification jstring="[para(574(a,1),569(a,1,2))].">
      <j1 rule="para" parents="574 569"/>
    </justification>
  </clause>

  <clause id="599">
    <literal><![CDATA[
      f(f(x,y),f(f(y,x),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(597)
    ]]></attribute>
    <justification jstring="[para(577(a,1),553(a,1,2,1,1)),demod(576(7))].">
      <j1 rule="para" parents="577 553"/>
      <j2 rule="demod" parents="576"/>
    </justification>
  </clause>

  <clause id="600">
    <literal><![CDATA[
      f(x,f(x,f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(598)
    ]]></attribute>
    <justification jstring="[para(577(a,1),554(a,1,2,1,1)),demod(576(7))].">
      <j1 rule="para" parents="577 554"/>
      <j2 rule="demod" parents="576"/>
    </justification>
  </clause>

  <clause id="601">
    <literal><![CDATA[
      f(x,f(x,f(x,y))) = f(x,y)
    ]]></literal>
    <attribute><![CDATA[
      label(599)
    ]]></attribute>
    <justification jstring="[para(577(a,1),555(a,1,2,1,1)),demod(576(7))].">
      <j1 rule="para" parents="577 555"/>
      <j2 rule="demod" parents="576"/>
    </justification>
  </clause>

  <clause id="603">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(x,f(y,x)),f(f(y,x),z)))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(601)
    ]]></attribute>
    <justification jstring="[para(577(a,1),550(a,1,2,2,1,1,1)),demod(576(6),576(5))].">
      <j1 rule="para" parents="577 550"/>
      <j2 rule="demod" parents="576 576"/>
    </justification>
  </clause>

  <clause id="608">
    <literal><![CDATA[
      f(f(x,f(y,f(y,y))),f(z,f(z,z))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(613)
    ]]></attribute>
    <justification jstring="[para(577(a,2),576(a,1,1))].">
      <j1 rule="para" parents="577 576"/>
    </justification>
  </clause>

  <clause id="609">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(z,z)),f(f(f(f(f(y,x),y),y),y),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(606)
    ]]></attribute>
    <justification jstring="[para(579(a,1),519(a,1,2,1)),demod(540(7),540(8))].">
      <j1 rule="para" parents="579 519"/>
      <j2 rule="demod" parents="540 540"/>
    </justification>
  </clause>

  <clause id="610">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(f(f(f(y,z),y),y),y),f(y,u))),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(607)
    ]]></attribute>
    <justification jstring="[para(579(a,1),520(a,1,1,1)),demod(540(6),540(7))].">
      <j1 rule="para" parents="579 520"/>
      <j2 rule="demod" parents="540 540"/>
    </justification>
  </clause>

  <clause id="618">
    <literal><![CDATA[
      f(f(f(x,y),f(x,y)),f(x,x)) = f(f(x,y),f(f(x,y),f(x,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(637)
    ]]></attribute>
    <justification jstring="[para(574(a,1),570(a,1,2))].">
      <j1 rule="para" parents="574 570"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      f(f(x,y),f(f(f(x,f(x,y)),x),f(x,f(x,y)))) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(616)
    ]]></attribute>
    <justification jstring="[para(601(a,1),599(a,1,1))].">
      <j1 rule="para" parents="601 599"/>
    </justification>
  </clause>

  <clause id="628">
    <literal><![CDATA[
      f(f(f(x,y),x),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(620)
    ]]></attribute>
    <justification jstring="[para(590(a,1),549(a,1,2,1,1,1)),demod(590(6),593(6),590(9),583(9),590(9))].">
      <j1 rule="para" parents="590 549"/>
      <j2 rule="demod" parents="590 593 590 583 590"/>
    </justification>
  </clause>

  <clause id="629">
    <literal><![CDATA[
      f(f(x,y),x) = f(x,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(623)
    ]]></attribute>
    <justification jstring="[back_demod(624),demod(628(7))].">
      <j1 rule="back_demod" parents="624"/>
      <j2 rule="demod" parents="628"/>
    </justification>
  </clause>

  <clause id="632">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,f(y,z)),f(y,u))),f(y,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(626)
    ]]></attribute>
    <justification jstring="[back_demod(610),demod(629(4),629(5),601(5),629(4))].">
      <j1 rule="back_demod" parents="610"/>
      <j2 rule="demod" parents="629 629 601 629"/>
    </justification>
  </clause>

  <clause id="633">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(z,z)),f(f(y,f(y,x)),f(u,y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(627)
    ]]></attribute>
    <justification jstring="[back_demod(609),demod(629(5),629(6),601(6),629(5))].">
      <j1 rule="back_demod" parents="609"/>
      <j2 rule="demod" parents="629 629 601 629"/>
    </justification>
  </clause>

  <clause id="634">
    <literal><![CDATA[
      f(f(x,y),y) = f(y,f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(628B)
    ]]></attribute>
    <justification jstring="[para(494(a,1),629(a,1,1)),demod(494(14)),flip(a)].">
      <j1 rule="para" parents="494 629"/>
      <j2 rule="demod" parents="494"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="635">
    <literal><![CDATA[
      f(f(x,y),f(y,f(x,y))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(629)
    ]]></attribute>
    <justification jstring="[para(629(a,1),557(a,1,2,1,1)),demod(634(2),634(5),629(8),600(8),629(5),600(5),634(4))].">
      <j1 rule="para" parents="629 557"/>
      <j2 rule="demod" parents="634 634 629 600 629 600 634"/>
    </justification>
  </clause>

  <clause id="636">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),x),f(f(y,f(x,y)),f(y,z)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(630)
    ]]></attribute>
    <justification jstring="[para(629(a,1),552(a,1,2,2,1,1,1)),demod(629(6),601(6),634(5))].">
      <j1 rule="para" parents="629 552"/>
      <j2 rule="demod" parents="629 601 634"/>
    </justification>
  </clause>

  <clause id="638">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(u,f(u,u)),f(f(z,f(f(y,x),z)),y)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(632)
    ]]></attribute>
    <justification jstring="[back_demod(589),demod(634(8))].">
      <j1 rule="back_demod" parents="589"/>
      <j2 rule="demod" parents="634"/>
    </justification>
  </clause>

  <clause id="639">
    <literal><![CDATA[
      f(x,f(f(y,x),f(f(f(y,f(x,y)),f(y,x)),f(z,f(y,x))))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(633)
    ]]></attribute>
    <justification jstring="[back_demod(567),demod(634(3))].">
      <j1 rule="back_demod" parents="567"/>
      <j2 rule="demod" parents="634"/>
    </justification>
  </clause>

  <clause id="648">
    <literal><![CDATA[
      f(f(x,f(y,x)),f(x,f(f(y,x),f(x,z)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(640)
    ]]></attribute>
    <justification jstring="[para(635(a,1),603(a,1,2,1)),demod(635(8),629(5),600(5),635(7),635(11))].">
      <j1 rule="para" parents="635 603"/>
      <j2 rule="demod" parents="635 629 600 635 635"/>
    </justification>
  </clause>

  <clause id="655">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,z),f(z,u))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(644)
    ]]></attribute>
    <justification jstring="[para(600(a,1),632(a,1,1,2,1))].">
      <j1 rule="para" parents="600 632"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),f(f(y,z),f(u,z))),f(z,z)) = z
    ]]></literal>
    <attribute><![CDATA[
      label(646)
    ]]></attribute>
    <justification jstring="[para(517(a,1),655(a,1,1,2,2))].">
      <j1 rule="para" parents="517 655"/>
    </justification>
  </clause>

  <clause id="660">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),y) = f(f(u,f(u,u)),f(f(x,y),f(z,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(649)
    ]]></attribute>
    <justification jstring="[para(657(a,1),540(a,1,2)),demod(581(13))].">
      <j1 rule="para" parents="657 540"/>
      <j2 rule="demod" parents="581"/>
    </justification>
  </clause>

  <clause id="663">
    <literal><![CDATA[
      f(f(x,f(x,y)),f(y,x)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(652A)
    ]]></attribute>
    <justification jstring="[para(633(a,1),583(a,1)),flip(a)].">
      <j1 rule="para" parents="633 583"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="664">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(x,z))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(658)
    ]]></attribute>
    <justification jstring="[para(655(a,1),618(a,1,1,1)),demod(655(8),581(14),655(13),655(13),655(13))].">
      <j1 rule="para" parents="655 618"/>
      <j2 rule="demod" parents="655 581 655 655 655"/>
    </justification>
  </clause>

  <clause id="667">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(z,f(z,z)))) = f(x,f(x,x))
    ]]></literal>
    <attribute><![CDATA[
      label(660)
    ]]></attribute>
    <justification jstring="[para(579(a,1),664(a,1,2,2))].">
      <j1 rule="para" parents="579 664"/>
    </justification>
  </clause>

  <clause id="670">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),y) = f(y,f(f(x,y),f(z,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(664B)
    ]]></attribute>
    <justification jstring="[para(660(a,1),517(a,1,2,1)),demod(590(11)),flip(a)].">
      <j1 rule="para" parents="660 517"/>
      <j2 rule="demod" parents="590"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="672">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(z,y),f(x,y)))) = f(f(z,y),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(666)
    ]]></attribute>
    <justification jstring="[para(660(a,2),583(a,1,2)),demod(670(5))].">
      <j1 rule="para" parents="660 583"/>
      <j2 rule="demod" parents="670"/>
    </justification>
  </clause>

  <clause id="673">
    <literal><![CDATA[
      f(x,f(f(x,y),f(f(z,f(x,y)),x))) = f(f(z,f(x,y)),x)
    ]]></literal>
    <attribute><![CDATA[
      label(669)
    ]]></attribute>
    <justification jstring="[para(540(a,1),672(a,1,1)),demod(540(6),540(11))].">
      <j1 rule="para" parents="540 672"/>
      <j2 rule="demod" parents="540 540"/>
    </justification>
  </clause>

  <clause id="674">
    <literal><![CDATA[
      f(f(x,y),f(x,f(f(z,x),f(x,y)))) = f(f(z,x),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(667)
    ]]></attribute>
    <justification jstring="[para(545(a,1),672(a,1,1)),demod(545(6),545(11))].">
      <j1 rule="para" parents="545 672"/>
      <j2 rule="demod" parents="545 545"/>
    </justification>
  </clause>

  <clause id="675">
    <literal><![CDATA[
      f(f(x,y),f(y,f(f(y,z),f(x,y)))) = f(f(y,z),f(x,y))
    ]]></literal>
    <attribute><![CDATA[
      label(668)
    ]]></attribute>
    <justification jstring="[para(545(a,1),672(a,1,2,2,1)),demod(545(10))].">
      <j1 rule="para" parents="545 672"/>
      <j2 rule="demod" parents="545"/>
    </justification>
  </clause>

  <clause id="676">
    <literal><![CDATA[
      f(x,f(f(x,y),f(y,x))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(670)
    ]]></attribute>
    <justification jstring="[para(547(a,1),673(a,1,2,2,1)),demod(547(7))].">
      <j1 rule="para" parents="547 673"/>
      <j2 rule="demod" parents="547"/>
    </justification>
  </clause>

  <clause id="677">
    <literal><![CDATA[
      f(f(x,y),f(y,x)) = f(f(y,x),f(y,x))
    ]]></literal>
    <attribute><![CDATA[
      label(671A)
    ]]></attribute>
    <justification jstring="[para(676(a,1),675(a,1,2)),flip(a)].">
      <j1 rule="para" parents="676 675"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="678">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(z,y),f(y,z))) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(672)
    ]]></attribute>
    <justification jstring="[para(677(a,2),504(a,1,2))].">
      <j1 rule="para" parents="677 504"/>
    </justification>
  </clause>

  <clause id="679">
    <literal><![CDATA[
      f(f(x,f(y,x)),f(x,y)) = x
    ]]></literal>
    <attribute><![CDATA[
      label(673)
    ]]></attribute>
    <justification jstring="[para(677(a,1),648(a,1,2,2)),demod(518(6))].">
      <j1 rule="para" parents="677 648"/>
      <j2 rule="demod" parents="518"/>
    </justification>
  </clause>

  <clause id="680">
    <literal><![CDATA[
      f(x,y) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(933C)
    ]]></attribute>
    <justification jstring="[para(677(a,1),677(a,1,1)),demod(540(7),678(8))].">
      <j1 rule="para" parents="677 677"/>
      <j2 rule="demod" parents="540 678"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      f(x,f(f(y,x),f(y,f(z,f(y,x))))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(676)
    ]]></attribute>
    <justification jstring="[back_demod(639),demod(679(5))].">
      <j1 rule="back_demod" parents="639"/>
      <j2 rule="demod" parents="679"/>
    </justification>
  </clause>

  <clause id="692">
    <literal><![CDATA[
      f(f(f(x,f(x,x)),y),z) = f(z,f(y,y))
    ]]></literal>
    <attribute><![CDATA[
      label(679)
    ]]></attribute>
    <justification jstring="[para(680(a,1),592(a,1))].">
      <j1 rule="para" parents="680 592"/>
    </justification>
  </clause>

  <clause id="703">
    <literal><![CDATA[
      f(f(x,f(y,z)),f(f(y,z),f(z,y))) = f(z,y)
    ]]></literal>
    <attribute><![CDATA[
      label(686)
    ]]></attribute>
    <justification jstring="[para(680(a,1),678(a,1,1,2))].">
      <j1 rule="para" parents="680 678"/>
    </justification>
  </clause>

  <clause id="706">
    <literal><![CDATA[
      f(x,f(f(y,z),f(z,x))) = f(z,x)
    ]]></literal>
    <attribute><![CDATA[
      label(689)
    ]]></attribute>
    <justification jstring="[para(674(a,1),682(a,1,2))].">
      <j1 rule="para" parents="674 682"/>
    </justification>
  </clause>

  <clause id="713">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(y,f(y,x))),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(695)
    ]]></attribute>
    <justification jstring="[para(663(a,1),706(a,1,2,2)),demod(663(10))].">
      <j1 rule="para" parents="663 706"/>
      <j2 rule="demod" parents="663"/>
    </justification>
  </clause>

  <clause id="714">
    <literal><![CDATA[
      f(x,f(f(y,z),f(x,z))) = f(z,x)
    ]]></literal>
    <attribute><![CDATA[
      label(696)
    ]]></attribute>
    <justification jstring="[para(680(a,1),706(a,1,2,2))].">
      <j1 rule="para" parents="680 706"/>
    </justification>
  </clause>

  <clause id="715">
    <literal><![CDATA[
      f(x,f(f(y,x),f(z,y))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(697)
    ]]></attribute>
    <justification jstring="[para(680(a,1),706(a,1,2))].">
      <j1 rule="para" parents="680 706"/>
    </justification>
  </clause>

  <clause id="721">
    <literal><![CDATA[
      f(f(x,y),z) = f(z,f(y,f(z,f(x,y))))
    ]]></literal>
    <attribute><![CDATA[
      label(703A)
    ]]></attribute>
    <justification jstring="[para(547(a,1),714(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="547 714"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="722">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),z) = f(y,z)
    ]]></literal>
    <attribute><![CDATA[
      label(704A)
    ]]></attribute>
    <justification jstring="[para(714(a,1),680(a,1)),flip(a)].">
      <j1 rule="para" parents="714 680"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="723">
    <literal><![CDATA[
      f(x,f(f(x,y),f(z,y))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(705)
    ]]></attribute>
    <justification jstring="[para(680(a,1),714(a,1,2))].">
      <j1 rule="para" parents="680 714"/>
    </justification>
  </clause>

  <clause id="725">
    <literal><![CDATA[
      f(x,f(f(f(y,z),x),z)) = f(f(y,z),x)
    ]]></literal>
    <attribute><![CDATA[
      label(708)
    ]]></attribute>
    <justification jstring="[para(547(a,1),715(a,1,2,2))].">
      <j1 rule="para" parents="547 715"/>
    </justification>
  </clause>

  <clause id="741">
    <literal><![CDATA[
      f(f(f(x,y),f(z,y)),x) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(725)
    ]]></attribute>
    <justification jstring="[para(680(a,1),722(a,1,1))].">
      <j1 rule="para" parents="680 722"/>
    </justification>
  </clause>

  <clause id="743">
    <literal><![CDATA[
      f(x,f(f(x,f(y,z)),z)) = f(f(y,z),x)
    ]]></literal>
    <attribute><![CDATA[
      label(726)
    ]]></attribute>
    <justification jstring="[para(547(a,1),723(a,1,2,2))].">
      <j1 rule="para" parents="547 723"/>
    </justification>
  </clause>

  <clause id="744">
    <literal><![CDATA[
      f(f(f(x,f(y,z)),z),x) = f(f(y,z),x)
    ]]></literal>
    <attribute><![CDATA[
      label(728)
    ]]></attribute>
    <justification jstring="[para(547(a,1),741(a,1,1,2))].">
      <j1 rule="para" parents="547 741"/>
    </justification>
  </clause>

  <clause id="749">
    <literal><![CDATA[
      f(x,f(y,f(x,f(z,y)))) = f(x,f(z,y))
    ]]></literal>
    <attribute><![CDATA[
      label(731)
    ]]></attribute>
    <justification jstring="[para(721(a,1),680(a,1))].">
      <j1 rule="para" parents="721 680"/>
    </justification>
  </clause>

  <clause id="750">
    <literal><![CDATA[
      f(f(f(f(x,f(x,x)),y),z),u) = f(u,f(z,f(y,y)))
    ]]></literal>
    <attribute><![CDATA[
      label(732)
    ]]></attribute>
    <justification jstring="[para(692(a,2),721(a,1,1)),demod(749(11))].">
      <j1 rule="para" parents="692 721"/>
      <j2 rule="demod" parents="749"/>
    </justification>
  </clause>

  <clause id="758">
    <literal><![CDATA[
      f(f(x,f(y,f(z,x))),y) = f(y,f(z,x))
    ]]></literal>
    <attribute><![CDATA[
      label(736A)
    ]]></attribute>
    <justification jstring="[para(749(a,1),680(a,1)),flip(a)].">
      <j1 rule="para" parents="749 680"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="765">
    <literal><![CDATA[
      f(f(f(x,y),f(y,x)),f(z,f(x,y))) = f(y,x)
    ]]></literal>
    <attribute><![CDATA[
      label(740A)
    ]]></attribute>
    <justification jstring="[para(703(a,1),680(a,1)),flip(a)].">
      <j1 rule="para" parents="703 680"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="768">
    <literal><![CDATA[
      f(x,f(f(f(y,z),x),f(u,f(z,y)))) = f(f(y,z),x)
    ]]></literal>
    <attribute><![CDATA[
      label(743)
    ]]></attribute>
    <justification jstring="[para(765(a,1),725(a,1,2,1,1)),demod(765(12))].">
      <j1 rule="para" parents="765 725"/>
      <j2 rule="demod" parents="765"/>
    </justification>
  </clause>

  <clause id="771">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(f(y,x),z)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(759)
    ]]></attribute>
    <justification jstring="[para(638(a,1),749(a,1,2,2)),demod(587(9),638(18))].">
      <j1 rule="para" parents="638 749"/>
      <j2 rule="demod" parents="587 638"/>
    </justification>
  </clause>

  <clause id="772">
    <literal><![CDATA[
      f(f(f(x,f(f(y,z),x)),y),f(z,y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(760)
    ]]></attribute>
    <justification jstring="[para(638(a,1),758(a,1,1,2)),demod(587(8),638(18))].">
      <j1 rule="para" parents="638 758"/>
      <j2 rule="demod" parents="587 638"/>
    </justification>
  </clause>

  <clause id="773">
    <literal><![CDATA[
      f(x,f(f(y,f(x,y)),f(x,z))) = f(x,z)
    ]]></literal>
    <attribute><![CDATA[
      label(761)
    ]]></attribute>
    <justification jstring="[para(515(a,1),771(a,1,2,1,2,1)),demod(540(3))].">
      <j1 rule="para" parents="515 771"/>
      <j2 rule="demod" parents="540"/>
    </justification>
  </clause>

  <clause id="777">
    <literal><![CDATA[
      f(f(x,y),f(f(z,f(f(u,f(f(y,x),u)),y)),y)) = y
    ]]></literal>
    <attribute><![CDATA[
      label(763)
    ]]></attribute>
    <justification jstring="[para(771(a,1),714(a,1,2,2)),demod(772(14))].">
      <j1 rule="para" parents="771 714"/>
      <j2 rule="demod" parents="772"/>
    </justification>
  </clause>

  <clause id="780">
    <literal><![CDATA[
      f(f(x,x),f(f(y,x),f(f(x,x),z))) = f(f(x,x),z)
    ]]></literal>
    <attribute><![CDATA[
      label(765)
    ]]></attribute>
    <justification jstring="[para(667(a,1),773(a,1,2,1,2)),demod(608(8))].">
      <j1 rule="para" parents="667 773"/>
      <j2 rule="demod" parents="608"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      f(x,f(f(y,f(y,y)),f(f(z,f(x,x)),f(f(x,x),u)))) = f(x,x)
    ]]></literal>
    <attribute><![CDATA[
      label(748)
    ]]></attribute>
    <justification jstring="[para(750(a,2),655(a,1)),demod(504(5))].">
      <j1 rule="para" parents="750 655"/>
      <j2 rule="demod" parents="504"/>
    </justification>
  </clause>

  <clause id="793">
    <literal><![CDATA[
      f(x,f(f(y,f(x,x)),f(x,z))) = f(x,z)
    ]]></literal>
    <attribute><![CDATA[
      label(772)
    ]]></attribute>
    <justification jstring="[para(504(a,1),780(a,1,1)),demod(540(5),540(8))].">
      <j1 rule="para" parents="504 780"/>
      <j2 rule="demod" parents="540 540"/>
    </justification>
  </clause>

  <clause id="795">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),y),f(z,x))) = f(f(x,x),y)
    ]]></literal>
    <attribute><![CDATA[
      label(774)
    ]]></attribute>
    <justification jstring="[para(680(a,1),780(a,1,2))].">
      <j1 rule="para" parents="680 780"/>
    </justification>
  </clause>

  <clause id="796">
    <literal><![CDATA[
      f(x,f(f(f(x,x),y),f(x,z))) = f(x,z)
    ]]></literal>
    <attribute><![CDATA[
      label(775)
    ]]></attribute>
    <justification jstring="[para(680(a,1),793(a,1,2,1))].">
      <j1 rule="para" parents="680 793"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      f(x,f(f(f(x,x),y),f(z,x))) = f(z,x)
    ]]></literal>
    <attribute><![CDATA[
      label(776)
    ]]></attribute>
    <justification jstring="[para(600(a,1),796(a,1,2,2)),demod(600(8))].">
      <j1 rule="para" parents="600 796"/>
      <j2 rule="demod" parents="600"/>
    </justification>
  </clause>

  <clause id="800">
    <literal><![CDATA[
      f(f(f(f(x,x),y),f(z,x)),x) = f(z,x)
    ]]></literal>
    <attribute><![CDATA[
      label(779A)
    ]]></attribute>
    <justification jstring="[para(797(a,1),680(a,1)),flip(a)].">
      <j1 rule="para" parents="797 680"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="804">
    <literal><![CDATA[
      f(f(x,y),f(f(f(y,y),z),f(f(y,f(x,y)),f(y,u)))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(781)
    ]]></attribute>
    <justification jstring="[para(800(a,1),636(a,1,1)),demod(795(7),800(8))].">
      <j1 rule="para" parents="800 636"/>
      <j2 rule="demod" parents="795 800"/>
    </justification>
  </clause>

  <clause id="818">
    <literal><![CDATA[
      f(f(x,x),f(f(f(x,x),y),f(f(z,f(z,z)),f(x,u)))) = x
    ]]></literal>
    <attribute><![CDATA[
      label(783)
    ]]></attribute>
    <justification jstring="[para(579(a,1),