<?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>DE-cs.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13121 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:34 2006
The command was "/home/mccune/bin/prover9 -f cs.in DE.in".
]]></heading>

<proof number="1" length="81" max_count="25">

<comments><![CDATA[
% Proof 1 at 0.14 (+ 0.00) seconds: E.
% Length of proof is 81.
% Level of proof is 22.
% Maximum clause weight is 25.
% Given clauses 132.
]]></comments>

  <clause id="5">
    <literal><![CDATA[
      (x @ y) * (z @ y) = x * z @ y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

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

  <clause id="10">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x * y @ z = (x @ z) * (y @ z)
    ]]></literal>
    <justification jstring="[copy(5),flip(a)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      A * B * C * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1)),demod(9(4))].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * z != u * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(7,a,10,a)].">
      <j1 rule="hyper" parents="7 10"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1)),demod(9(2),9(5)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="demod" parents="9 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * y * z * (z @ x * y) = z * x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x * y * z * (y @ x) * (z @ x) = y * z * x
    ]]></literal>
    <justification jstring="[para(9(a,1),10(a,1,2)),demod(11(2),9(8))].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="demod" parents="11 9"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      (x @ y) * (z @ y) * ((z @ x) @ y) = (z @ y) * (x @ y)
    ]]></literal>
    <justification jstring="[para(10(a,1),11(a,1,1)),demod(11(2),11(7)),flip(a)].">
      <j1 rule="para" parents="10 11"/>
      <j2 rule="demod" parents="11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      A * B * C * B * A * x != B * A * C * A * B * x
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[ur(8,b,12,a),demod(9(10),9(9),9(8),9(7),9(20),9(19),9(18),9(17))].">
      <j1 rule="ur" parents="8 12"/>
      <j2 rule="demod" parents="9 9 9 9 9 9 9 9"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(15(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="15 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(15(a,1),9(a,1,1)),flip(a)].">
      <j1 rule="para" parents="15 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x * y * z != x * u * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ y) = z
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      x * y * z != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = z
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="10 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      x * (y @ z) != y * z
    ]]></literal>
    <literal><![CDATA[
      z * y = x
    ]]></literal>
    <justification jstring="[para(10(a,1),14(a,1)),flip(a)].">
      <j1 rule="para" parents="10 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(13,a,30,a)].">
      <j1 rule="hyper" parents="13 30"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      z @ z = x
    ]]></literal>
    <justification jstring="[para(49(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="49 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="56">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(49(a,1),10(a,1,2)),demod(49(5))].">
      <j1 rule="para" parents="49 10"/>
      <j2 rule="demod" parents="49"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[para(11(a,1),49(a,1,1)),demod(9(6))].">
      <j1 rule="para" parents="11 49"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      x * y * z != u * x * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ x * y) = z
    ]]></literal>
    <justification jstring="[para(9(a,1),16(a,1))].">
      <j1 rule="para" parents="9 16"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(27,a,56,a),flip(a)].">
      <j1 rule="hyper" parents="27 56"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      (u @ u) @ y = z
    ]]></literal>
    <justification jstring="[para(56(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="56 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      (x @ x) @ y * z = (y @ y * z) * (z @ y * z)
    ]]></literal>
    <justification jstring="[para(70(a,2),11(a,1))].">
      <j1 rule="para" parents="70 11"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(70(a,2),49(a,1,1))].">
      <j1 rule="para" parents="70 49"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(70(a,2),56(a,1,2,1))].">
      <j1 rule="para" parents="70 56"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(16,a,89,a)].">
      <j1 rule="hyper" parents="16 89"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      (z @ z) @ u = x
    ]]></literal>
    <justification jstring="[para(89(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="89 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(89(a,1),16(a,2)),demod(89(8))].">
      <j1 rule="para" parents="89 16"/>
      <j2 rule="demod" parents="89"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(70(a,2),89(a,1,1,1))].">
      <j1 rule="para" parents="70 89"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(55,a,89,a),flip(a)].">
      <j1 rule="hyper" parents="55 89"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(55,a,49,a)].">
      <j1 rule="hyper" parents="55 49"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(115)].">
      <j1 rule="new_symbol" parents="115"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_demod(114),demod(117(1),117(3))].">
      <j1 rule="back_demod" parents="114"/>
      <j2 rule="demod" parents="117 117"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_demod(109),demod(117(1),119(2),119(2))].">
      <j1 rule="back_demod" parents="109"/>
      <j2 rule="demod" parents="117 119 119"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_demod(107),demod(117(3),119(4),119(4))].">
      <j1 rule="back_demod" parents="107"/>
      <j2 rule="demod" parents="117 119 119"/>
    </justification>
  </clause>

  <clause id="127">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      c_0 = x
    ]]></literal>
    <justification jstring="[back_demod(101),demod(117(3),119(4))].">
      <j1 rule="back_demod" parents="101"/>
      <j2 rule="demod" parents="117 119"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_demod(100),demod(117(1),119(2),117(4),119(5))].">
      <j1 rule="back_demod" parents="100"/>
      <j2 rule="demod" parents="117 119 117 119"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_demod(90),demod(117(1),119(2),119(2))].">
      <j1 rule="back_demod" parents="90"/>
      <j2 rule="demod" parents="117 119 119"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_demod(86),demod(117(1),119(3)),flip(a)].">
      <j1 rule="back_demod" parents="86"/>
      <j2 rule="demod" parents="117 119"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = z
    ]]></literal>
    <justification jstring="[back_demod(79),demod(117(5),119(6))].">
      <j1 rule="back_demod" parents="79"/>
      <j2 rule="demod" parents="117 119"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_demod(128),demod(129(5))].">
      <j1 rule="back_demod" parents="128"/>
      <j2 rule="demod" parents="129"/>
    </justification>
  </clause>

  <clause id="168">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(124,a,135,a),flip(a)].">
      <j1 rule="hyper" parents="124 135"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      x * y * (y @ y * x) = y * x
    ]]></literal>
    <justification jstring="[hyper(7,a,20,a)].">
      <j1 rule="hyper" parents="7 20"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      A * B * C * B * A * (B @ C * A) != B * A * B * C * A
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(20(a,1),23(a,2,2,2))].">
      <j1 rule="para" parents="20 23"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      (x @ y) * ((x @ y) @ y) = x @ y
    ]]></literal>
    <justification jstring="[para(117(a,1),22(a,1,1)),demod(123(6),117(6),129(7))].">
      <j1 rule="para" parents="117 22"/>
      <j2 rule="demod" parents="123 117 129"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      x @ y = c_0
    ]]></literal>
    <justification jstring="[para(10(a,1),134(a,1)),flip(b)].">
      <j1 rule="para" parents="10 134"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="258">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(16,a,170,a),flip(a)].">
      <j1 rule="hyper" parents="16 170"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="270">
    <literal><![CDATA[
      (x @ y * x) * (y @ y * x) = c_0
    ]]></literal>
    <justification jstring="[para(131(a,1),170(a,1,2,2,2)),demod(168(8),129(6),131(10))].">
      <j1 rule="para" parents="131 170"/>
      <j2 rule="demod" parents="168 129 131"/>
    </justification>
  </clause>

  <clause id="273">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(210(a,1),10(a,1,2)),flip(a)].">
      <j1 rule="para" parents="210 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="292">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      z = y
    ]]></literal>
    <justification jstring="[para(123(a,1),38(a,1,2)),demod(129(3),168(5),129(5))].">
      <j1 rule="para" parents="123 38"/>
      <j2 rule="demod" parents="129 168 129"/>
    </justification>
  </clause>

  <clause id="297">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(222,a,273,a(flip))].">
      <j1 rule="hyper" parents="222 273"/>
    </justification>
  </clause>

  <clause id="309">
    <literal><![CDATA[
      x * (y @ x) != x
    ]]></literal>
    <literal><![CDATA[
      y @ x = c_0
    ]]></literal>
    <justification jstring="[para(273(a,1),127(a,1)),flip(b)].">
      <j1 rule="para" parents="273 127"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="322">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[hyper(39,a,22,a)].">
      <j1 rule="hyper" parents="39 22"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = c_0
    ]]></literal>
    <justification jstring="[para(129(a,1),39(a,1,2))].">
      <j1 rule="para" parents="129 39"/>
    </justification>
  </clause>

  <clause id="337">
    <literal><![CDATA[
      (x @ y * x) @ (y @ y * x) = c_0
    ]]></literal>
    <justification jstring="[para(131(a,1),326(a,1)),demod(270(6)),xx(a)].">
      <j1 rule="para" parents="131 326"/>
      <j2 rule="demod" parents="270"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="407">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(292,a,258,a),flip(a)].">
      <j1 rule="hyper" parents="292 258"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="427">
    <literal><![CDATA[
      (x @ y * x) @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[back_demod(337),demod(407(4))].">
      <j1 rule="back_demod" parents="337"/>
      <j2 rule="demod" parents="407"/>
    </justification>
  </clause>

  <clause id="429">
    <literal><![CDATA[
      (x @ y * x) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[back_demod(270),demod(407(4))].">
      <j1 rule="back_demod" parents="270"/>
      <j2 rule="demod" parents="407"/>
    </justification>
  </clause>

  <clause id="447">
    <literal><![CDATA[
      (x @ y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[back_demod(59),demod(407(2))].">
      <j1 rule="back_demod" parents="59"/>
      <j2 rule="demod" parents="407"/>
    </justification>
  </clause>

  <clause id="453">
    <literal><![CDATA[
      x @ y * (y @ x) = x @ y * x
    ]]></literal>
    <justification jstring="[para(10(a,1),407(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 407"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="494">
    <literal><![CDATA[
      (x @ y * x) * z * (y @ x) = z * (z @ (y @ x))
    ]]></literal>
    <justification jstring="[para(427(a,1),21(a,1,2,2,2,1)),demod(123(7),447(8)),flip(a)].">
      <j1 rule="para" parents="427 21"/>
      <j2 rule="demod" parents="123 447"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="540">
    <literal><![CDATA[
      x * (y @ z) * ((y @ x) @ (z @ x)) = (y @ z) * x
    ]]></literal>
    <justification jstring="[para(322(a,2),10(a,1,2,2))].">
      <j1 rule="para" parents="322 10"/>
    </justification>
  </clause>

  <clause id="563">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = c_0
    ]]></literal>
    <justification jstring="[para(297(a,1),322(a,1,2)),demod(168(4)),flip(a)].">
      <j1 rule="para" parents="297 322"/>
      <j2 rule="demod" parents="168"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="568">
    <literal><![CDATA[
      (x @ y) @ z = c_0
    ]]></literal>
    <justification jstring="[para(322(a,2),309(a,1,2)),demod(563(3),129(2)),xx(a)].">
      <j1 rule="para" parents="322 309"/>
      <j2 rule="demod" parents="563 129"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="571">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(322(a,2),429(a,1,1)),demod(563(7),123(4))].">
      <j1 rule="para" parents="322 429"/>
      <j2 rule="demod" parents="563 123"/>
    </justification>
  </clause>

  <clause id="580">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_demod(540),demod(571(4),129(3)),flip(a)].">
      <j1 rule="back_demod" parents="540"/>
      <j2 rule="demod" parents="571 129"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      (x @ y * x) * z * (y @ x) = z
    ]]></literal>
    <justification jstring="[back_demod(494),demod(571(7),129(7))].">
      <j1 rule="back_demod" parents="494"/>
      <j2 rule="demod" parents="571 129"/>
    </justification>
  </clause>

  <clause id="606">
    <literal><![CDATA[
      x * (x @ y * x) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(64,a,19,a),demod(580(5))].">
      <j1 rule="hyper" parents="64 19"/>
      <j2 rule="demod" parents="580"/>
    </justification>
  </clause>

  <clause id="736">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[para(10(a,1),453(a,2,2)),demod(453(5),606(5),11(5),453(3),568(6),129(4),11(6),407(4),568(6),129(5))].">
      <j1 rule="para" parents="10 453"/>
      <j2 rule="demod" parents="453 606 11 453 568 129 11 407 568 129"/>
    </justification>
  </clause>

  <clause id="773">
    <literal><![CDATA[
      (x @ y) * z * (y @ x) = z
    ]]></literal>
    <justification jstring="[back_demod(593),demod(736(2))].">
      <j1 rule="back_demod" parents="593"/>
      <j2 rule="demod" parents="736"/>
    </justification>
  </clause>

  <clause id="794">
    <literal><![CDATA[
      x @ y * z * x = x @ y * z
    ]]></literal>
    <justification jstring="[para(9(a,1),736(a,1,2))].">
      <j1 rule="para" parents="9 736"/>
    </justification>
  </clause>

  <clause id="836">
    <literal><![CDATA[
      (x @ y * z) * u * (y @ x) * (z @ x) = u
    ]]></literal>
    <justification jstring="[para(11(a,1),773(a,1,2,2))].">
      <j1 rule="para" parents="11 773"/>
    </justification>
  </clause>

  <clause id="928">
    <literal><![CDATA[
      x @ y * z = x @ z * y
    ]]></literal>
    <justification jstring="[para(19(a,1),794(a,1,2)),demod(11(5),568(4),794(4),123(4),11(6),568(5),123(6))].">
      <j1 rule="para" parents="19 794"/>
      <j2 rule="demod" parents="11 568 794 123 11 568 123"/>
    </justification>
  </clause>

  <clause id="950">
    <literal><![CDATA[
      A * B * C * B * A * (B @ A * C) != B * A * B * C * A
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[back_demod(176),demod(928(10))].">
      <j1 rule="back_demod" parents="176"/>
      <j2 rule="demod" parents="928"/>
    </justification>
  </clause>

  <clause id="995">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[ur(46,b,950,a(flip)),demod(11(24),11(25),117(21),11(24),580(26,R),123(27),9(27),9(26),9(25),9(24),9(23),836(22),10(10),9(18),9(17),9(16)),xx(a)].">
      <j1 rule="ur" parents="46 950"/>
      <j2 rule="demod" parents="11 11 117 11 580 123 9 9 9 9 9 836 10 9 9 9"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
