<?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="1">

<source>identity.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13014 was started by mccune on cleo.thornwood,
Mon Jun 19 16:40:06 2006
The command was "/home/mccune/bin/prover9 -f identity.in".
]]></heading>

<proof number="1" length="87" max_count="51">

<comments><![CDATA[
% Proof 1 at 3.92 (+ 0.02) seconds.
% Length of proof is 87.
% Level of proof is 18.
% Maximum clause weight is 51.
% Given clauses 92.
]]></comments>

  <clause id="6">
    <literal><![CDATA[
      (x * e) * x = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      x * (x * y) = y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      (x * y) * (z * u) = (x * z) * (y * u)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      ((x * x) * x) * x = e
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      e * e = e
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1)),demod(7(5)),flip(a)].">
      <j1 rule="para" parents="6 6"/>
      <j2 rule="demod" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      ((x * y) * x) * (e * y) = x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),6(a,1))].">
      <j1 rule="para" parents="8 6"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x * e) * y) * (x * z) = x * (y * z)
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,1)),flip(a)].">
      <j1 rule="para" parents="6 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (x * (y * e)) * (z * y) = (x * z) * y
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="6 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      (x * y) * ((x * z) * (y * u)) = z * u
    ]]></literal>
    <justification jstring="[para(8(a,1),7(a,1,2))].">
      <j1 rule="para" parents="8 7"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x * y) * ((x * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(7(a,1),8(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x * y) * (z * (y * u)) = (x * z) * u
    ]]></literal>
    <justification jstring="[para(7(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      ((x * x) * x) * e = x
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1,2))].">
      <j1 rule="para" parents="9 7"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      (((x * y) * x) * ((x * y) * y)) * e = x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),20(a,1,1))].">
      <j1 rule="para" parents="8 20"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (x * y) * (e * (y * x)) = x
    ]]></literal>
    <justification jstring="[para(7(a,1),12(a,1,1,1)),demod(7(7))].">
      <j1 rule="para" parents="7 12"/>
      <j2 rule="demod" parents="7"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      (x * y) * x = e * (y * x)
    ]]></literal>
    <justification jstring="[para(34(a,1),7(a,1,2))].">
      <j1 rule="para" parents="34 7"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x * (e * (x * x)) = e * (x * x)
    ]]></literal>
    <justification jstring="[para(9(a,1),34(a,1,1)),demod(40(4),7(8),40(6))].">
      <j1 rule="para" parents="9 34"/>
      <j2 rule="demod" parents="40 7 40"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      e * ((x * x) * e) = x
    ]]></literal>
    <justification jstring="[para(9(a,1),34(a,1,2,2)),demod(40(2),45(4),11(6),40(5))].">
      <j1 rule="para" parents="9 34"/>
      <j2 rule="demod" parents="40 45 11 40"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      (x * x) * (e * x) = e
    ]]></literal>
    <justification jstring="[para(20(a,1),34(a,1,2,2)),demod(40(3),7(5))].">
      <j1 rule="para" parents="20 34"/>
      <j2 rule="demod" parents="40 7"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      ((e * (x * y)) * ((y * x) * x)) * e = y * x
    ]]></literal>
    <justification jstring="[back_demod(32),demod(40(2))].">
      <j1 rule="back_demod" parents="32"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      (e * (x * x)) * x = e
    ]]></literal>
    <justification jstring="[back_demod(9),demod(40(2))].">
      <j1 rule="back_demod" parents="9"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      (x * x) * e = e * x
    ]]></literal>
    <justification jstring="[para(46(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="46 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      ((x * e) * y) * z = x * (y * (x * z))
    ]]></literal>
    <justification jstring="[para(7(a,1),13(a,1,2))].">
      <j1 rule="para" parents="7 13"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      (e * ((x * (y * e)) * y)) * (y * (x * (y * z))) = y * (x * z)
    ]]></literal>
    <justification jstring="[para(13(a,1),13(a,2)),demod(65(5),40(5),65(10))].">
      <j1 rule="para" parents="13 13"/>
      <j2 rule="demod" parents="65 40 65"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      (x * e) * (x * x) = e
    ]]></literal>
    <justification jstring="[para(47(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="47 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      (e * x) * (x * x) = x * x
    ]]></literal>
    <justification jstring="[para(47(a,1),34(a,1,1)),demod(7(8))].">
      <j1 rule="para" parents="47 34"/>
      <j2 rule="demod" parents="7"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      ((x * x) * y) * (e * z) = (e * x) * (y * z)
    ]]></literal>
    <justification jstring="[para(62(a,1),8(a,1,1)),flip(a)].">
      <j1 rule="para" parents="62 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      (x * e) * e = x * x
    ]]></literal>
    <justification jstring="[para(77(a,1),7(a,1,2))].">
      <j1 rule="para" parents="77 7"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      (x * (e * (x * e))) * (x * (e * x)) = e
    ]]></literal>
    <justification jstring="[para(14(a,2),47(a,1)),demod(40(4))].">
      <j1 rule="para" parents="14 47"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      (x * x) * y = x * ((y * e) * (x * (e * y)))
    ]]></literal>
    <justification jstring="[para(86(a,1),14(a,2,1)),demod(65(8)),flip(a)].">
      <j1 rule="para" parents="86 14"/>
      <j2 rule="demod" parents="65"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      (x * x) * (x * (y * (x * (e * z)))) = y * z
    ]]></literal>
    <justification jstring="[para(86(a,1),15(a,1,1)),demod(65(7))].">
      <j1 rule="para" parents="86 15"/>
      <j2 rule="demod" parents="65"/>
    </justification>
  </clause>

  <clause id="189">
    <literal><![CDATA[
      (e * (x * y)) * ((x * x) * (y * y)) = (x * y) * (x * y)
    ]]></literal>
    <justification jstring="[para(8(a,1),79(a,1,2))].">
      <j1 rule="para" parents="8 79"/>
    </justification>
  </clause>

  <clause id="211">
    <literal><![CDATA[
      e * (x * y) = z * (x * (e * (z * y)))
    ]]></literal>
    <justification jstring="[para(34(a,1),16(a,1,2)),demod(40(2))].">
      <j1 rule="para" parents="34 16"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="216">
    <literal><![CDATA[
      (x * y) * e = x * (y * (e * x))
    ]]></literal>
    <justification jstring="[para(47(a,1),16(a,1,2))].">
      <j1 rule="para" parents="47 16"/>
    </justification>
  </clause>

  <clause id="243">
    <literal><![CDATA[
      ((x * y) * z) * (((x * (u * e)) * (y * u)) * v) = u * (z * v)
    ]]></literal>
    <justification jstring="[para(14(a,2),16(a,1,2,1))].">
      <j1 rule="para" parents="14 16"/>
    </justification>
  </clause>

  <clause id="259">
    <literal><![CDATA[
      (e * (x * y)) * (((y * x) * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(40(a,1),16(a,1,1))].">
      <j1 rule="para" parents="40 16"/>
    </justification>
  </clause>

  <clause id="275">
    <literal><![CDATA[
      ((x * y) * ((x * e) * z)) * ((y * z) * (y * z)) = (y * z) * (y * z)
    ]]></literal>
    <justification jstring="[para(16(a,2),79(a,1,1))].">
      <j1 rule="para" parents="16 79"/>
    </justification>
  </clause>

  <clause id="316">
    <literal><![CDATA[
      x * (y * (x * y)) = y * x
    ]]></literal>
    <justification jstring="[back_demod(55),demod(216(8),7(10),259(8))].">
      <j1 rule="back_demod" parents="55"/>
      <j2 rule="demod" parents="216 7 259"/>
    </justification>
  </clause>

  <clause id="318">
    <literal><![CDATA[
      (e * x) * (x * (e * x)) = e
    ]]></literal>
    <justification jstring="[back_demod(112),demod(316(5))].">
      <j1 rule="back_demod" parents="112"/>
      <j2 rule="demod" parents="316"/>
    </justification>
  </clause>

  <clause id="331">
    <literal><![CDATA[
      x * (y * x) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(316(a,1),7(a,1,2))].">
      <j1 rule="para" parents="316 7"/>
    </justification>
  </clause>

  <clause id="372">
    <literal><![CDATA[
      x * ((e * x) * x) = e
    ]]></literal>
    <justification jstring="[back_demod(318),demod(331(6))].">
      <j1 rule="back_demod" parents="318"/>
      <j2 rule="demod" parents="331"/>
    </justification>
  </clause>

  <clause id="376">
    <literal><![CDATA[
      (e * x) * x = x * e
    ]]></literal>
    <justification jstring="[para(372(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="372 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="384">
    <literal><![CDATA[
      x * (y * (e * x)) = z * (y * (x * (z * (e * x))))
    ]]></literal>
    <justification jstring="[para(372(a,1),16(a,1,2)),demod(216(3),376(9),216(7))].">
      <j1 rule="para" parents="372 16"/>
      <j2 rule="demod" parents="216 376 216"/>
    </justification>
  </clause>

  <clause id="401">
    <literal><![CDATA[
      (e * x) * y = e * (x * (e * y))
    ]]></literal>
    <justification jstring="[para(11(a,1),17(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="407">
    <literal><![CDATA[
      ((x * y) * z) * u = x * (z * (e * ((y * x) * (e * u))))
    ]]></literal>
    <justification jstring="[para(34(a,1),17(a,1,1)),demod(401(4)),flip(a)].">
      <j1 rule="para" parents="34 17"/>
      <j2 rule="demod" parents="401"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="410">
    <literal><![CDATA[
      e * (x * (y * z)) = y * (x * (e * z))
    ]]></literal>
    <justification jstring="[para(61(a,1),17(a,1,1)),demod(401(8),401(11),82(12),401(10),401(13),7(13),7(12))].">
      <j1 rule="para" parents="61 17"/>
      <j2 rule="demod" parents="401 401 82 401 401 7 7"/>
    </justification>
  </clause>

  <clause id="438">
    <literal><![CDATA[
      (x * y) * (z * (y * x)) = e * (z * x)
    ]]></literal>
    <justification jstring="[para(17(a,2),40(a,1))].">
      <j1 rule="para" parents="17 40"/>
    </justification>
  </clause>

  <clause id="446">
    <literal><![CDATA[
      x * (y * (e * ((z * x) * (e * (x * ((z * (u * v)) * (e * ((u * x) * (e * (y * w)))))))))) = v * w
    ]]></literal>
    <justification jstring="[para(17(a,2),15(a,1,2,1)),demod(407(8),407(14))].">
      <j1 rule="para" parents="17 15"/>
      <j2 rule="demod" parents="407 407"/>
    </justification>
  </clause>

  <clause id="448">
    <literal><![CDATA[
      (x * y) * (z * (y * ((x * u) * (z * v)))) = u * v
    ]]></literal>
    <justification jstring="[para(17(a,2),15(a,1))].">
      <j1 rule="para" parents="17 15"/>
    </justification>
  </clause>

  <clause id="520">
    <literal><![CDATA[
      e * ((x * y) * (e * (y * (z * (e * ((x * y) * (e * u))))))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_demod(259),demod(407(6),401(12))].">
      <j1 rule="back_demod" parents="259"/>
      <j2 rule="demod" parents="407 401"/>
    </justification>
  </clause>

  <clause id="525">
    <literal><![CDATA[
      e * ((x * y) * (e * ((x * x) * (y * y)))) = (x * y) * (x * y)
    ]]></literal>
    <justification jstring="[back_demod(189),demod(401(7))].">
      <j1 rule="back_demod" parents="189"/>
      <j2 rule="demod" parents="401"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      e * (x * (y * (e * z))) = y * (x * z)
    ]]></literal>
    <justification jstring="[back_demod(74),demod(401(10),407(11),7(12),407(9),401(11),7(11),7(8),7(8),7(5),7(4))].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="401 407 7 407 401 7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="568">
    <literal><![CDATA[
      x * (y * (e * ((z * y) * (e * ((z * x) * (z * x)))))) = (z * x) * (z * x)
    ]]></literal>
    <justification jstring="[back_demod(275),demod(407(9),407(13),7(14),401(11),7(14),7(13))].">
      <j1 rule="back_demod" parents="275"/>
      <j2 rule="demod" parents="407 407 7 401 7 7"/>
    </justification>
  </clause>

  <clause id="581">
    <literal><![CDATA[
      x * (y * (e * (x * (z * (e * u))))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_demod(243),demod(407(8),407(10),7(11),401(8),7(11),407(13),448(15),7(7))].">
      <j1 rule="back_demod" parents="243"/>
      <j2 rule="demod" parents="407 407 7 401 7 407 448 7"/>
    </justification>
  </clause>

  <clause id="628">
    <literal><![CDATA[
      c4 * (c3 * (e * ((c2 * c4) * (e * c1)))) != c1 * (c3 * (e * ((c2 * c1) * (e * c4))))
    ]]></literal>
    <justification jstring="[back_demod(10),demod(407(7),407(20))].">
      <j1 rule="back_demod" parents="10"/>
      <j2 rule="demod" parents="407 407"/>
    </justification>
  </clause>

  <clause id="630">
    <literal><![CDATA[
      (x * x) * y = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_demod(142),demod(438(8))].">
      <j1 rule="back_demod" parents="142"/>
      <j2 rule="demod" parents="438"/>
    </justification>
  </clause>

  <clause id="631">
    <literal><![CDATA[
      x * (y * (e * ((z * u) * ((z * x) * (e * (y * v)))))) = u * v
    ]]></literal>
    <justification jstring="[back_demod(446),demod(448(16))].">
      <j1 rule="back_demod" parents="446"/>
      <j2 rule="demod" parents="448"/>
    </justification>
  </clause>

  <clause id="644">
    <literal><![CDATA[
      e * ((x * y) * (e * (x * (e * (x * (y * y)))))) = (x * y) * (x * y)
    ]]></literal>
    <justification jstring="[back_demod(525),demod(630(6))].">
      <j1 rule="back_demod" parents="525"/>
      <j2 rule="demod" parents="630"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      x * (e * (y * (x * (e * z)))) = y * z
    ]]></literal>
    <justification jstring="[back_demod(175),demod(630(7),7(7))].">
      <j1 rule="back_demod" parents="175"/>
      <j2 rule="demod" parents="630 7"/>
    </justification>
  </clause>

  <clause id="667">
    <literal><![CDATA[
      c1 * (c3 * (e * ((c2 * c1) * (e * c4)))) != c4 * (c3 * (e * ((c2 * e) * (c4 * c1))))
    ]]></literal>
    <justification jstring="[para(8(a,1),628(a,1,2,2,2)),flip(a)].">
      <j1 rule="para" parents="8 628"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="681">
    <literal><![CDATA[
      c4 * (c3 * (e * ((c2 * e) * (c4 * c1)))) != c1 * (c3 * (e * ((c2 * e) * (c1 * c4))))
    ]]></literal>
    <justification jstring="[para(8(a,1),667(a,1,2,2,2)),flip(a)].">
      <j1 rule="para" parents="8 667"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="716">
    <literal><![CDATA[
      e * (x * y) = x * (y * (e * (y * x)))
    ]]></literal>
    <justification jstring="[para(331(a,1),17(a,1)),demod(630(2),40(7)),flip(a)].">
      <j1 rule="para" parents="331 17"/>
      <j2 rule="demod" parents="630 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="742">
    <literal><![CDATA[
      (x * y) * z = e * (x * (e * (x * (e * (y * (e * (x * (e * (x * (e * z))))))))))
    ]]></literal>
    <justification jstring="[para(45(a,1),17(a,1,1)),demod(401(7),630(8),401(13),630(14)),flip(a)].">
      <j1 rule="para" parents="45 17"/>
      <j2 rule="demod" parents="401 630 401 630"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * (e * (x * (e * (x * (e * (x * y))))))))))) = x * (e * (x * (y * y)))
    ]]></literal>
    <justification jstring="[back_demod(644),demod(742(10),7(15),7(12),7(11),7(8),45(8),7(8),7(9),742(8)),flip(a)].">
      <j1 rule="back_demod" parents="644"/>
      <j2 rule="demod" parents="742 7 7 7 7 45 7 7 742"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="749">
    <literal><![CDATA[
      x * (y * (z * (e * (z * (e * (y * (e * (x * x)))))))) = z * (e * (z * (x * x)))
    ]]></literal>
    <justification jstring="[back_demod(568),demod(742(6),748(21),742(10),7(15),7(12),7(11),7(8),7(14),742(15),748(30))].">
      <j1 rule="back_demod" parents="568"/>
      <j2 rule="demod" parents="742 748 742 7 7 7 7 7 742 748"/>
    </justification>
  </clause>

  <clause id="795">
    <literal><![CDATA[
      c4 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c4 * c1)))))))))) != c1 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c1 * c4))))))))))
    ]]></literal>
    <justification jstring="[back_demod(681),demod(742(10),7(23),7(25),742(33),7(46),7(48))].">
      <j1 rule="back_demod" parents="681"/>
      <j2 rule="demod" parents="742 7 7 742 7 7"/>
    </justification>
  </clause>

  <clause id="819">
    <literal><![CDATA[
      x * (y * (z * (e * (z * (e * (u * (x * (e * (z * (e * (z * (y * v)))))))))))) = u * v
    ]]></literal>
    <justification jstring="[back_demod(631),demod(742(7),7(12),742(19),7(24),7(21),7(20),7(17),7(16),7(19))].">
      <j1 rule="back_demod" parents="631"/>
      <j2 rule="demod" parents="742 7 742 7 7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="900">
    <literal><![CDATA[
      x * (e * (x * (e * (y * (e * (x * (e * (x * (y * (z * (x * (e * (x * (e * (y * (e * (x * (e * (x * u))))))))))))))))))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_demod(520),demod(742(8),7(13),7(20),742(20),7(25),7(32))].">
      <j1 rule="back_demod" parents="520"/>
      <j2 rule="demod" parents="742 7 7 742 7 7"/>
    </justification>
  </clause>

  <clause id="1144">
    <literal><![CDATA[
      e * (x * (y * e)) = y * (x * e)
    ]]></literal>
    <justification jstring="[para(11(a,1),410(a,2,2,2))].">
      <j1 rule="para" parents="11 410"/>
    </justification>
  </clause>

  <clause id="1155">
    <literal><![CDATA[
      e * (x * (y * x)) = x * (y * (e * y))
    ]]></literal>
    <justification jstring="[para(331(a,1),410(a,1,2))].">
      <j1 rule="para" parents="331 410"/>
    </justification>
  </clause>

  <clause id="1165">
    <literal><![CDATA[
      e * (x * (y * z)) = u * (x * (y * (u * (e * z))))
    ]]></literal>
    <justification jstring="[para(410(a,1),211(a,2,2,2))].">
      <j1 rule="para" parents="410 211"/>
    </justification>
  </clause>

  <clause id="1195">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * (y * (e * (x * e))))))))))) = x * (y * (x * e))
    ]]></literal>
    <justification jstring="[para(316(a,1),384(a,2,2,2,2)),demod(742(8),7(18),742(24),7(29),45(28),7(28),7(26),7(23),316(24))].">
      <j1 rule="para" parents="316 384"/>
      <j2 rule="demod" parents="742 7 742 7 45 7 7 7 316"/>
    </justification>
  </clause>

  <clause id="1220">
    <literal><![CDATA[
      x * (e * (y * (x * (y * e)))) = e * y
    ]]></literal>
    <justification jstring="[para(1144(a,1),384(a,1,2)),demod(11(5),742(5),316(11),7(9),7(8),45(6),7(6),7(3),742(11),7(21),1195(22)),flip(a)].">
      <j1 rule="para" parents="1144 384"/>
      <j2 rule="demod" parents="11 742 316 7 7 45 7 7 742 7 1195"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1314">
    <literal><![CDATA[
      e * (x * (y * (e * (y * x)))) = x * y
    ]]></literal>
    <justification jstring="[para(536(a,2),316(a,1))].">
      <j1 rule="para" parents="536 316"/>
    </justification>
  </clause>

  <clause id="1325">
    <literal><![CDATA[
      e * (x * (y * (e * (z * (e * (x * u)))))) = y * (e * (z * u))
    ]]></literal>
    <justification jstring="[para(211(a,2),536(a,2,2))].">
      <j1 rule="para" parents="211 536"/>
    </justification>
  </clause>

  <clause id="1329">
    <literal><![CDATA[
      c4 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c4 * c1)))))))))) != e * (c3 * (c1 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c1 * c4))))))))))))
    ]]></literal>
    <justification jstring="[para(536(a,2),795(a,2))].">
      <j1 rule="para" parents="536 795"/>
    </justification>
  </clause>

  <clause id="1909">
    <literal><![CDATA[
      e * (x * (y * (z * (x * (e * u))))) = y * (z * u)
    ]]></literal>
    <justification jstring="[para(657(a,1),410(a,2,2))].">
      <j1 rule="para" parents="657 410"/>
    </justification>
  </clause>

  <clause id="1928">
    <literal><![CDATA[
      x * (e * (x * (e * (y * (z * (e * (x * (e * (x * u))))))))) = y * (z * u)
    ]]></literal>
    <justification jstring="[back_demod(900),demod(1909(21),1909(17),7(12))].">
      <j1 rule="back_demod" parents="900"/>
      <j2 rule="demod" parents="1909 1909 7"/>
    </justification>
  </clause>

  <clause id="1936">
    <literal><![CDATA[
      x * (y * (z * (x * (y * u)))) = z * u
    ]]></literal>
    <justification jstring="[back_demod(819),demod(1928(15))].">
      <j1 rule="back_demod" parents="819"/>
      <j2 rule="demod" parents="1928"/>
    </justification>
  </clause>

  <clause id="1954">
    <literal><![CDATA[
      x * (e * (x * (e * (x * (e * (x * (e * y))))))) = e * (x * (e * (x * (e * (x * y)))))
    ]]></literal>
    <justification jstring="[para(1144(a,1),716(a,2,2,2)),demod(742(4),7(14),7(16),742(19),1220(25),7(21),7(20))].">
      <j1 rule="para" parents="1144 716"/>
      <j2 rule="demod" parents="742 7 7 742 1220 7 7"/>
    </justification>
  </clause>

  <clause id="2131">
    <literal><![CDATA[
      c4 * (c3 * (e * (c2 * (e * (c2 * (e * (c2 * (c4 * c1)))))))) != e * (c3 * (c1 * (c2 * (e * (c2 * (e * (c2 * (c1 * c4))))))))
    ]]></literal>
    <justification jstring="[back_demod(1329),demod(1954(21),1954(42),7(39))].">
      <j1 rule="back_demod" parents="1329"/>
      <j2 rule="demod" parents="1954 1954 7"/>
    </justification>
  </clause>

  <clause id="2335">
    <literal><![CDATA[
      c4 * (c3 * (e * (c2 * (e * (c2 * (c4 * (c2 * (e * c1)))))))) != e * (c3 * (c1 * (c2 * (e * (c2 * (e * (c2 * (c1 * c4))))))))
    ]]></literal>
    <justification jstring="[para(581(a,2),2131(a,1,2,2,2,2,2,2,2)),demod(1936(19))].">
      <j1 rule="para" parents="581 2131"/>
      <j2 rule="demod" parents="1936"/>
    </justification>
  </clause>

  <clause id="2713">
    <literal><![CDATA[
      c4 * (c3 * (e * (c2 * (c4 * (c2 * (e * (c2 * (e * c1)))))))) != e * (c3 * (c1 * (c2 * (e * (c2 * (e * (c2 * (c1 * c4))))))))
    ]]></literal>
    <justification jstring="[para(581(a,2),2335(a,1,2,2,2,2,2)),demod(1936(21))].">
      <j1 rule="para" parents="581 2335"/>
      <j2 rule="demod" parents="1936"/>
    </justification>
  </clause>

  <clause id="2886">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * y))))))) = x * (e * (x * (e * (x * (e * y)))))
    ]]></literal>
    <justification jstring="[para(1155(a,2),716(a,2,2,2,2)),demod(742(5),742(15),7(23),7(21),7(19),7(16),7(15),7(13),7(11),7(13),7(18),7(15),742(14),742(24),7(32),7(30),7(28),7(25),7(24),7(22),7(20)),flip(a)].">
      <j1 rule="para" parents="1155 716"/>
      <j2 rule="demod" parents="742 742 7 7 7 7 7 7 7 7 7 7 742 742 7 7 7 7 7 7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3877">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (y * (e * (y * (e * x))))))))) = y * (e * (y * x))
    ]]></literal>
    <justification jstring="[para(1155(a,2),749(a,1,2,2,2,2,2,2,2,2)),demod(7(14),742(13),7(18),45(17),7(17),742(14),7(23),7(21),45(19),7(19),7(17),7(15),45(13),7(13),7(10),7(10),742(14),1325(19),742(18),7(26),7(24),7(22),7(19),7(18),7(16),7(14),742(23),316(28),7(26),11(25),742(26),7(34),7(32),7(30),7(27),11(26),11(25),11(24),11(23),11(22),11(21),316(22),7(20))].">
      <j1 rule="para" parents="1155 749"/>
      <j2 rule="demod" parents="7 742 7 45 7 742 7 7 45 7 7 7 45 7 7 7 742 1325 742 7 7 7 7 7 7 7 742 316 7 11 742 7 7 7 7 11 11 11 11 11 11 316 7"/>
    </justification>
  </clause>

  <clause id="3896">
    <literal><![CDATA[
      c4 * (c3 * (c1 * (c4 * (e * (c2 * (e * (c2 * (e * (c2 * (c1 * c4)))))))))) != e * (c3 * (c1 * (c2 * (e * (c2 * (e * (c2 * (c1 * c4))))))))
    ]]></literal>
    <justification jstring="[para(716(a,1),2713(a,1,2,2)),demod(742(28),742(41),7(49),742(50),7(58),7(56),7(54),742(51),3877(61),1314(49),742(42),7(51),7(49),7(49),7(47),7(45),1936(43),7(38),7(36),7(34),7(36),7(37),742(34),2886(46),742(43),742(56),7(64),7(62),7(60),742(57),1954(68),7(65),742(62),7(71),7(69),7(69),7(67),7(65),1936(63),1936(57),7(51),7(49),7(47),1936(45),7(40),1909(38),1909(32),7(25))].">
      <j1 rule="para" parents="716 2713"/>
      <j2 rule="demod" parents="742 742 7 742 7 7 7 742 3877 1314 742 7 7 7 7 7 1936 7 7 7 7 7 742 2886 742 742 7 7 7 742 1954 7 742 7 7 7 7 7 1936 1936 7 7 7 1936 7 1909 1909 7"/>
    </justification>
  </clause>

  <clause id="3897">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(3896,a,1165,a(flip))].">
      <j1 rule="resolve" parents="3896 1165"/>
    </justification>
  </clause>

</proof>

</proofs>
