<?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>identity2.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 10670 was started by mccune on cleo.thornwood,
Sat Aug 12 20:57:50 2006
The command was "/home/mccune/bin/prover9 -f identity2.in".
]]></heading>

<proof number="1" length="91" max_count="65">

<comments><![CDATA[
% Proof 1 at 9.22 (+ 0.02) seconds.
% Length of proof is 91.
% Level of proof is 16.
% Maximum clause weight is 65.
% Given clauses 121.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      ((x * y) * z) * u = ((u * y) * z) * x
    ]]></literal>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="2">
    <literal><![CDATA[
      (x * e) * x = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3">
    <literal><![CDATA[
      x * (x * y) = y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      (x * y) * (z * u) = (x * z) * (y * u)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5">
    <literal><![CDATA[
      ((x * x) * x) * x = e
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

  <clause id="10">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      e * e = e
    ]]></literal>
    <justification jstring="[para(2(a,1),2(a,1,1)),rewrite(3(5)),flip(a)].">
      <j1 rule="para" parents="2 2"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      ((x * y) * x) * (e * y) = x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),2(a,1))].">
      <j1 rule="para" parents="4 2"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x * e) * y) * (x * z) = x * (y * z)
    ]]></literal>
    <justification jstring="[para(2(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="2 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="15">
    <literal><![CDATA[
      (x * y) * ((x * z) * (y * u)) = z * u
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,2))].">
      <j1 rule="para" parents="4 3"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x * y) * ((x * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x * y) * (z * (y * u)) = (x * z) * u
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,2)),flip(a)].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="28">
    <literal><![CDATA[
      (x / y) * x = y
    ]]></literal>
    <justification jstring="[para(8(a,1),3(a,1,2))].">
      <j1 rule="para" parents="8 3"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x / (y * x) = y
    ]]></literal>
    <justification jstring="[para(3(a,1),9(a,1,1))].">
      <j1 rule="para" parents="3 9"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      e / x = (x * x) * x
    ]]></literal>
    <justification jstring="[para(5(a,1),9(a,1,1))].">
      <j1 rule="para" parents="5 9"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      (x * y) * (e * (y * x)) = x
    ]]></literal>
    <justification jstring="[para(3(a,1),12(a,1,1,1)),rewrite(3(7))].">
      <j1 rule="para" parents="3 12"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (x * (x / y)) * (e * y) = x
    ]]></literal>
    <justification jstring="[para(8(a,1),12(a,1,1,1)),rewrite(8(7))].">
      <j1 rule="para" parents="8 12"/>
      <j2 rule="rewrite" parents="8"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x * ((x * x) * x) = (x * x) * x
    ]]></literal>
    <justification jstring="[para(28(a,1),2(a,1,1)),rewrite(34(2),34(5))].">
      <j1 rule="para" parents="28 2"/>
      <j2 rule="rewrite" parents="34 34"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      x / e = (x * x) * x
    ]]></literal>
    <justification jstring="[para(5(a,1),32(a,1,2))].">
      <j1 rule="para" parents="5 32"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      ((x * e) * y) * z = x * (y * (x * z))
    ]]></literal>
    <justification jstring="[para(3(a,1),13(a,1,2))].">
      <j1 rule="para" parents="3 13"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (x * (y * (x * z))) * ((x * u) * v) = (x * (y * u)) * (z * v)
    ]]></literal>
    <justification jstring="[para(13(a,1),4(a,1,1)),rewrite(56(8)),flip(a)].">
      <j1 rule="para" parents="13 4"/>
      <j2 rule="rewrite" parents="56"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      ((x * x) * x) * (y * x) = (x * y) * e
    ]]></literal>
    <justification jstring="[para(5(a,1),13(a,1,2)),rewrite(20(4)),flip(a)].">
      <j1 rule="para" parents="5 13"/>
      <j2 rule="rewrite" parents="20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      ((x * x) * x) * (y * z) = (x * y) * (((x * x) * x) * z)
    ]]></literal>
    <justification jstring="[para(8(a,1),13(a,1,1,1)),rewrite(52(3),52(7)),flip(a)].">
      <j1 rule="para" parents="8 13"/>
      <j2 rule="rewrite" parents="52 52"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x * ((x * e) * e) = x
    ]]></literal>
    <justification jstring="[para(34(a,1),32(a,1)),rewrite(56(8),3(5))].">
      <j1 rule="para" parents="34 32"/>
      <j2 rule="rewrite" parents="56 3"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      (x * e) * e = x * x
    ]]></literal>
    <justification jstring="[para(76(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="76 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      (x * y) * x = e * (y * x)
    ]]></literal>
    <justification jstring="[para(3(a,1),14(a,1,1)),flip(a)].">
      <j1 rule="para" parents="3 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      (e * (x * x)) * x = e
    ]]></literal>
    <justification jstring="[para(14(a,2),5(a,1,1)),rewrite(3(3))].">
      <j1 rule="para" parents="14 5"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != ((c1 * c2) * (c4 * e)) * (c3 * c4)
    ]]></literal>
    <justification jstring="[para(14(a,2),10(a,2))].">
      <j1 rule="para" parents="14 10"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      e * ((x * x) * e) = x
    ]]></literal>
    <justification jstring="[para(14(a,2),20(a,1,1)),rewrite(3(3),83(5))].">
      <j1 rule="para" parents="14 20"/>
      <j2 rule="rewrite" parents="3 83"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      (e * (x * x)) * (y * z) = (x * y) * ((e * (x * x)) * z)
    ]]></literal>
    <justification jstring="[back_rewrite(64),rewrite(83(2),83(8))].">
      <j1 rule="back_rewrite" parents="64"/>
      <j2 rule="rewrite" parents="83 83"/>
    </justification>
  </clause>

  <clause id="133">
    <literal><![CDATA[
      (e * (x * x)) * (y * x) = (x * y) * e
    ]]></literal>
    <justification jstring="[back_rewrite(62),rewrite(83(2))].">
      <j1 rule="back_rewrite" parents="62"/>
      <j2 rule="rewrite" parents="83"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      x * (e * (x * x)) = e * (x * x)
    ]]></literal>
    <justification jstring="[back_rewrite(46),rewrite(83(2),83(6))].">
      <j1 rule="back_rewrite" parents="46"/>
      <j2 rule="rewrite" parents="83 83"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      (x * x) * y = x * ((y * e) * (x * (e * y)))
    ]]></literal>
    <justification jstring="[para(77(a,1),14(a,2,1)),rewrite(56(8)),flip(a)].">
      <j1 rule="para" parents="77 14"/>
      <j2 rule="rewrite" parents="56"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      ((x * y) * (x * y)) * y = e * (x * e)
    ]]></literal>
    <justification jstring="[para(93(a,1),15(a,1,2)),rewrite(83(4)),flip(a)].">
      <j1 rule="para" parents="93 15"/>
      <j2 rule="rewrite" parents="83"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      (x * x) * e = e * x
    ]]></literal>
    <justification jstring="[para(114(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="114 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="209">
    <literal><![CDATA[
      x * (((y / x) * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(28(a,1),16(a,1,1))].">
      <j1 rule="para" parents="28 16"/>
    </justification>
  </clause>

  <clause id="238">
    <literal><![CDATA[
      (x * x) * (y * x) = e * (y * e)
    ]]></literal>
    <justification jstring="[para(93(a,1),16(a,1,2)),rewrite(83(4)),flip(a)].">
      <j1 rule="para" parents="93 16"/>
      <j2 rule="rewrite" parents="83"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != (x * c3) * ((x * ((c1 * c2) * (c4 * e))) * c4)
    ]]></literal>
    <justification jstring="[para(16(a,2),104(a,2))].">
      <j1 rule="para" parents="16 104"/>
    </justification>
  </clause>

  <clause id="283">
    <literal><![CDATA[
      (x * e) * (x * e) = e * x
    ]]></literal>
    <justification jstring="[para(190(a,1),14(a,2)),rewrite(11(3))].">
      <j1 rule="para" parents="190 14"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="288">
    <literal><![CDATA[
      (x * y) * (e * x) = x * (y * e)
    ]]></literal>
    <justification jstring="[para(190(a,1),16(a,1,2))].">
      <j1 rule="para" parents="190 16"/>
    </justification>
  </clause>

  <clause id="289">
    <literal><![CDATA[
      (x * (y * y)) * ((x * z) * e) = z * (e * y)
    ]]></literal>
    <justification jstring="[para(190(a,1),16(a,2,2))].">
      <j1 rule="para" parents="190 16"/>
    </justification>
  </clause>

  <clause id="298">
    <literal><![CDATA[
      ((x / y) * z) * u = x * (z * (y * u))
    ]]></literal>
    <justification jstring="[para(8(a,1),17(a,1,1)),flip(a)].">
      <j1 rule="para" parents="8 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="304">
    <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="305">
    <literal><![CDATA[
      x * (y * (z * u)) = z * (y * (x * u))
    ]]></literal>
    <justification jstring="[para(28(a,1),17(a,1,1)),rewrite(298(6))].">
      <j1 rule="para" parents="28 17"/>
      <j2 rule="rewrite" parents="298"/>
    </justification>
  </clause>

  <clause id="307">
    <literal><![CDATA[
      x * (y * (z * (x * (y * u)))) = z * u
    ]]></literal>
    <justification jstring="[para(28(a,1),17(a,2,1)),rewrite(298(5))].">
      <j1 rule="para" parents="28 17"/>
      <j2 rule="rewrite" parents="298"/>
    </justification>
  </clause>

  <clause id="392">
    <literal><![CDATA[
      x * (y * (z * (x * u))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_rewrite(209),rewrite(298(3))].">
      <j1 rule="back_rewrite" parents="209"/>
      <j2 rule="rewrite" parents="298"/>
    </justification>
  </clause>

  <clause id="398">
    <literal><![CDATA[
      (x * y) * e = e * ((x * x) * (e * (y * x)))
    ]]></literal>
    <justification jstring="[back_rewrite(133),rewrite(304(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="133"/>
      <j2 rule="rewrite" parents="304"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="400">
    <literal><![CDATA[
      (x * y) * (e * ((x * x) * (e * z))) = e * ((x * x) * (e * (y * z)))
    ]]></literal>
    <justification jstring="[back_rewrite(131),rewrite(304(5),304(12)),flip(a)].">
      <j1 rule="back_rewrite" parents="131"/>
      <j2 rule="rewrite" parents="304 304"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="453">
    <literal><![CDATA[
      ((x * y) * z) * u = x * (z * (e * ((y * x) * (e * u))))
    ]]></literal>
    <justification jstring="[para(38(a,1),17(a,1,1)),rewrite(304(4)),flip(a)].">
      <j1 rule="para" parents="38 17"/>
      <j2 rule="rewrite" parents="304"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="566">
    <literal><![CDATA[
      (x * c3) * ((x * ((c1 * c2) * (c4 * e))) * c4) != c4 * (c3 * (e * ((c2 * c4) * (e * c1))))
    ]]></literal>
    <justification jstring="[back_rewrite(253),rewrite(453(7)),flip(a)].">
      <j1 rule="back_rewrite" parents="253"/>
      <j2 rule="rewrite" parents="453"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="595">
    <literal><![CDATA[
      x * ((x * y) * (e * (y * (x * e)))) = e * (x * e)
    ]]></literal>
    <justification jstring="[back_rewrite(183),rewrite(453(4),288(6))].">
      <j1 rule="back_rewrite" parents="183"/>
      <j2 rule="rewrite" parents="453 288"/>
    </justification>
  </clause>

  <clause id="629">
    <literal><![CDATA[
      x * (y * (e * (z * (e * (x * (y * u)))))) = e * (z * (e * u))
    ]]></literal>
    <justification jstring="[para(44(a,1),15(a,1,2,1)),rewrite(453(6),28(3),304(12))].">
      <j1 rule="para" parents="44 15"/>
      <j2 rule="rewrite" parents="453 28 304"/>
    </justification>
  </clause>

  <clause id="854">
    <literal><![CDATA[
      (x * y) * (z * (y * x)) = e * (z * x)
    ]]></literal>
    <justification jstring="[para(83(a,1),17(a,2))].">
      <j1 rule="para" parents="83 17"/>
    </justification>
  </clause>

  <clause id="875">
    <literal><![CDATA[
      (x * x) * y = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_rewrite(160),rewrite(854(8))].">
      <j1 rule="back_rewrite" parents="160"/>
      <j2 rule="rewrite" parents="854"/>
    </justification>
  </clause>

  <clause id="888">
    <literal><![CDATA[
      (x * y) * (e * (x * (e * (x * (e * z))))) = e * (x * (e * (x * (e * (y * z)))))
    ]]></literal>
    <justification jstring="[back_rewrite(400),rewrite(875(6),875(16))].">
      <j1 rule="back_rewrite" parents="400"/>
      <j2 rule="rewrite" parents="875 875"/>
    </justification>
  </clause>

  <clause id="889">
    <literal><![CDATA[
      (x * y) * e = e * (x * (e * (x * (e * (y * x)))))
    ]]></literal>
    <justification jstring="[back_rewrite(398),rewrite(875(9))].">
      <j1 rule="back_rewrite" parents="398"/>
      <j2 rule="rewrite" parents="875"/>
    </justification>
  </clause>

  <clause id="893">
    <literal><![CDATA[
      e * (x * e) = y * (e * (y * (x * y)))
    ]]></literal>
    <justification jstring="[back_rewrite(238),rewrite(875(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="238"/>
      <j2 rule="rewrite" parents="875"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="914">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * (e * (y * (z * x)))))))) = z * (e * y)
    ]]></literal>
    <justification jstring="[back_rewrite(289),rewrite(889(5),888(12),875(6))].">
      <j1 rule="back_rewrite" parents="289"/>
      <j2 rule="rewrite" parents="889 888 875"/>
    </justification>
  </clause>

  <clause id="1079">
    <literal><![CDATA[
      e * (x * (y * e)) = y * (x * e)
    ]]></literal>
    <justification jstring="[para(283(a,1),16(a,1,2)),rewrite(288(4)),flip(a)].">
      <j1 rule="para" parents="283 16"/>
      <j2 rule="rewrite" parents="288"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1087">
    <literal><![CDATA[
      (x * y) * (e * (y * (x * e))) = e * x
    ]]></literal>
    <justification jstring="[para(283(a,1),17(a,2))].">
      <j1 rule="para" parents="283 17"/>
    </justification>
  </clause>

  <clause id="1102">
    <literal><![CDATA[
      e * (x * e) = x * (e * x)
    ]]></literal>
    <justification jstring="[back_rewrite(595),rewrite(1087(7)),flip(a)].">
      <j1 rule="back_rewrite" parents="595"/>
      <j2 rule="rewrite" parents="1087"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1239">
    <literal><![CDATA[
      (x * c3) * ((x * ((c1 * c2) * (c4 * e))) * c4) != c4 * (c3 * (e * ((c2 * e) * (c4 * c1))))
    ]]></literal>
    <justification jstring="[para(4(a,1),566(a,2,2,2,2))].">
      <j1 rule="para" parents="4 566"/>
    </justification>
  </clause>

  <clause id="1352">
    <literal><![CDATA[
      e * ((x * (e * y)) * z) = e * ((x * e) * (e * (y * z)))
    ]]></literal>
    <justification jstring="[para(11(a,1),57(a,1,2,1)),rewrite(304(8),3(8),304(12))].">
      <j1 rule="para" parents="11 57"/>
      <j2 rule="rewrite" parents="304 3 304"/>
    </justification>
  </clause>

  <clause id="1612">
    <literal><![CDATA[
      c1 * (c3 * (e * ((c4 * c1) * (e * ((c2 * e) * c4))))) != c4 * (c3 * (e * ((c2 * e) * (c4 * c1))))
    ]]></literal>
    <justification jstring="[para(15(a,1),1239(a,1,2,1)),rewrite(453(11))].">
      <j1 rule="para" parents="15 1239"/>
      <j2 rule="rewrite" parents="453"/>
    </justification>
  </clause>

  <clause id="1760">
    <literal><![CDATA[
      e * (x * (e * ((y * e) * (e * (y * (x * z)))))) = (y * e) * z
    ]]></literal>
    <justification jstring="[para(1102(a,1),15(a,1,2,1)),rewrite(304(8),1352(8))].">
      <j1 rule="para" parents="1102 15"/>
      <j2 rule="rewrite" parents="304 1352"/>
    </justification>
  </clause>

  <clause id="1763">
    <literal><![CDATA[
      (x * e) * (x * (e * (x * (e * (y * x))))) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(1102(a,2),15(a,1,2)),rewrite(889(6),3(13))].">
      <j1 rule="para" parents="1102 15"/>
      <j2 rule="rewrite" parents="889 3"/>
    </justification>
  </clause>

  <clause id="2347">
    <literal><![CDATA[
      (x * y) * z = e * (x * (e * (x * (e * (y * (e * (x * (e * (x * (e * z))))))))))
    ]]></literal>
    <justification jstring="[para(136(a,1),17(a,1,1)),rewrite(304(7),875(8),304(13),875(14)),flip(a)].">
      <j1 rule="para" parents="136 17"/>
      <j2 rule="rewrite" parents="304 875 304 875"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2718">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * (x * (e * (x * (e * (y * x))))))))))))) = y * (x * y)
    ]]></literal>
    <justification jstring="[back_rewrite(1763),rewrite(2347(10),3(20))].">
      <j1 rule="back_rewrite" parents="1763"/>
      <j2 rule="rewrite" parents="2347 3"/>
    </justification>
  </clause>

  <clause id="2719">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * y)))))))) = e * (z * (x * (e * (x * (e * (x * (e * (z * y))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(1760),rewrite(2347(9),3(14),3(11),3(13),3(15),2347(16),3(26)),flip(a)].">
      <j1 rule="back_rewrite" parents="1760"/>
      <j2 rule="rewrite" parents="2347 3 3 3 3 2347 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2808">
    <literal><![CDATA[
      c1 * (c3 * (c4 * (e * (c4 * (e * (c1 * (e * (c4 * (e * (c4 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * c4))))))))))))))))))) != c4 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c4 * c1))))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(1612),rewrite(2347(12),3(25),3(27),2347(24),3(43),2347(51),3(64),3(66))].">
      <j1 rule="back_rewrite" parents="1612"/>
      <j2 rule="rewrite" parents="2347 3 3 2347 3 2347 3 3"/>
    </justification>
  </clause>

  <clause id="3176">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * (x * (e * y)))))))))) = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_rewrite(875),rewrite(2347(2))].">
      <j1 rule="back_rewrite" parents="875"/>
      <j2 rule="rewrite" parents="2347"/>
    </justification>
  </clause>

  <clause id="3523">
    <literal><![CDATA[
      x * (y * x) = y * (x * y)
    ]]></literal>
    <justification jstring="[back_rewrite(2718),rewrite(3176(19),3(7),3(6))].">
      <j1 rule="back_rewrite" parents="2718"/>
      <j2 rule="rewrite" parents="3176 3 3"/>
    </justification>
  </clause>

  <clause id="3540">
    <literal><![CDATA[
      x * (y * (x * y)) = y * x
    ]]></literal>
    <justification jstring="[para(3523(a,1),3(a,1,2))].">
      <j1 rule="para" parents="3523 3"/>
    </justification>
  </clause>

  <clause id="3562">
    <literal><![CDATA[
      x * (y * (z * y)) = z * (x * (y * x))
    ]]></literal>
    <justification jstring="[para(3523(a,1),305(a,1,2)),flip(a)].">
      <j1 rule="para" parents="3523 305"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3587">
    <literal><![CDATA[
      x * (y * (z * (y * x))) = z * (x * y)
    ]]></literal>
    <justification jstring="[para(3540(a,1),305(a,1,2)),flip(a)].">
      <j1 rule="para" parents="3540 305"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3607">
    <literal><![CDATA[
      x * (e * (y * (x * e))) = y * e
    ]]></literal>
    <justification jstring="[para(11(a,1),307(a,1,2,2,2,2))].">
      <j1 rule="para" parents="11 307"/>
    </justification>
  </clause>

  <clause id="3619">
    <literal><![CDATA[
      x * (y * (z * (u * (v * (x * (u * w)))))) = z * (y * (v * w))
    ]]></literal>
    <justification jstring="[para(307(a,1),305(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="307 305"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3798">
    <literal><![CDATA[
      x * (y * (z * (x * (z * y)))) = y * z
    ]]></literal>
    <justification jstring="[para(392(a,2),3540(a,1))].">
      <j1 rule="para" parents="392 3540"/>
    </justification>
  </clause>

  <clause id="4640">
    <literal><![CDATA[
      x * (e * (y * (e * (y * (x * y))))) = e
    ]]></literal>
    <justification jstring="[para(893(a,1),3607(a,1,2,2)),rewrite(11(11))].">
      <j1 rule="para" parents="893 3607"/>
      <j2 rule="rewrite" parents="11"/>
    </justification>
  </clause>

  <clause id="4669">
    <literal><![CDATA[
      x * (y * (z * (e * (u * (e * (y * (x * v))))))) = z * (e * (u * (e * v)))
    ]]></literal>
    <justification jstring="[para(629(a,1),305(a,1,2)),flip(a)].">
      <j1 rule="para" parents="629 305"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4949">
    <literal><![CDATA[
      e * (x * (e * (y * (e * (y * (e * (y * (e * (y * (e * x)))))))))) = x * (e * (y * y))
    ]]></literal>
    <justification jstring="[para(893(a,1),3562(a,1,2,2)),rewrite(2347(8),4640(14),11(8),3(9),3540(7),3(5),2347(8),3(18)),flip(a)].">
      <j1 rule="para" parents="893 3562"/>
      <j2 rule="rewrite" parents="2347 4640 11 3 3540 3 2347 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4993">
    <literal><![CDATA[
      c1 * (c3 * (c4 * (e * (c4 * (e * (c1 * (c2 * c2))))))) != c4 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c4 * c1))))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(2808),rewrite(4949(32),3(17),3(14))].">
      <j1 rule="back_rewrite" parents="2808"/>
      <j2 rule="rewrite" parents="4949 3 3"/>
    </justification>
  </clause>

  <clause id="5429">
    <literal><![CDATA[
      x * (e * (x * (e * (x * (e * (x * (e * y))))))) = e * (x * (e * (x * (e * (x * y)))))
    ]]></literal>
    <justification jstring="[para(1079(a,1),3587(a,1,2,2)),rewrite(2347(7),3798(13),3(9),3(8),2347(13),3(23),3(25)),flip(a)].">
      <j1 rule="para" parents="1079 3587"/>
      <j2 rule="rewrite" parents="2347 3798 3 3 2347 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5601">
    <literal><![CDATA[
      c1 * (c3 * (c4 * (e * (c4 * (e * (c1 * (c2 * c2))))))) != c4 * (c3 * (e * (c2 * (e * (c2 * (e * (c2 * (c4 * c1))))))))
    ]]></literal>
    <justification jstring="[para(3587(a,2),4993(a,1,2,2,2,2,2,2)),rewrite(3(15),5429(38))].">
      <j1 rule="para" parents="3587 4993"/>
      <j2 rule="rewrite" parents="3 5429"/>
    </justification>
  </clause>

  <clause id="6020">
    <literal><![CDATA[
      e * (x * (y * (e * (y * (e * (y * (e * (x * z)))))))) = y * (e * (y * (e * (y * z))))
    ]]></literal>
    <justification jstring="[back_rewrite(2719),rewrite(5429(13),3(11)),flip(a)].">
      <j1 rule="back_rewrite" parents="2719"/>
      <j2 rule="rewrite" parents="5429 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6698">
    <literal><![CDATA[
      c4 * (c3 * (e * (c2 * (e * (c2 * (e * (c2 * (c4 * c1)))))))) != c1 * (c3 * (c4 * (x * (e * (x * (c1 * (c2 * (e * (c2 * (e * (c2 * (c4 * x))))))))))))
    ]]></literal>
    <justification jstring="[para(914(a,2),5601(a,1,2,2,2,2)),rewrite(2347(21),2347(34),3(42),5429(47),3(38),3(40),3(42),2347(39),5429(50),4669(42),4669(36),3(31),2347(28),307(37),307(31),3(31),6020(29),3(24),3(25)),flip(a)].">
      <j1 rule="para" parents="914 5601"/>
      <j2 rule="rewrite" parents="2347 2347 3 5429 3 3 3 2347 5429 4669 4669 3 2347 307 307 3 6020 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6699">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(6698,a,3619,a(flip))].">
      <j1 rule="resolve" parents="6698 3619"/>
    </justification>
  </clause>

</proof>

</proofs>
