<?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 April-2007, April 2007.
Process 26941 was started by mccune on cleo,
Fri Apr 13 09:16:37 2007
The command was "/home/mccune/bin/prover9 -f cs.in DE.in".
]]></heading>

<proof number="1" length="80" max_count="25">

<comments><![CDATA[
% Proof 1 at 0.17 (+ 0.01) seconds: E.
% Length of proof is 80.
% Level of proof is 20.
% Maximum clause weight is 25.
% Given clauses 143.
]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

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

  <clause id="6">
    <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="7" type="assumption">
    <literal><![CDATA[
      A * B * C * B * A != B * A * C * A * B
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1)),rewrite(3(4))].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x * y * z != u * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u
    ]]></literal>
    <justification jstring="[para(3(a,1),2(a,1))].">
      <j1 rule="para" parents="3 2"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x * y * (y @ z) != y * z
    ]]></literal>
    <literal><![CDATA[
      z = x
    ]]></literal>
    <justification jstring="[para(4(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="4 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,1)),rewrite(3(2),3(5)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x * y * z * (z @ x * y) = z * x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x @ y) * (z @ y) * ((z @ x) @ y) = (z @ y) * (x @ y)
    ]]></literal>
    <justification jstring="[para(4(a,1),6(a,1,1)),rewrite(6(2),6(7)),flip(a)].">
      <j1 rule="para" parents="4 6"/>
      <j2 rule="rewrite" parents="6 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      A * B * C * B * A * x != B * A * C * A * B * x
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[ur(2,b,7,a),rewrite(3(10),3(9),3(8),3(7),3(20),3(19),3(18),3(17))].">
      <j1 rule="ur" parents="2 7"/>
      <j2 rule="rewrite" parents="3 3 3 3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x * y * z != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = z
    ]]></literal>
    <justification jstring="[para(4(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="4 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="10 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x * (y @ z) != y * z
    ]]></literal>
    <literal><![CDATA[
      z * y = x
    ]]></literal>
    <justification jstring="[para(4(a,1),9(a,1)),flip(a)].">
      <j1 rule="para" parents="4 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(8,a,28,a)].">
      <j1 rule="hyper" parents="8 28"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      z @ z = x
    ]]></literal>
    <justification jstring="[para(44(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="44 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(44(a,1),4(a,1,2)),rewrite(44(5))].">
      <j1 rule="para" parents="44 4"/>
      <j2 rule="rewrite" parents="44"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[para(6(a,1),44(a,1,1)),rewrite(3(6))].">
      <j1 rule="para" parents="6 44"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(25,a,51,a),flip(a)].">
      <j1 rule="hyper" parents="25 51"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      (x @ x) @ y * z = (y @ y * z) * (z @ y * z)
    ]]></literal>
    <justification jstring="[para(58(a,2),6(a,1))].">
      <j1 rule="para" parents="58 6"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(58(a,2),44(a,1,1))].">
      <j1 rule="para" parents="58 44"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(58(a,2),51(a,1,2,1))].">
      <j1 rule="para" parents="58 51"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(11,a,75,a)].">
      <j1 rule="hyper" parents="11 75"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(58(a,2),75(a,1,1,1))].">
      <j1 rule="para" parents="58 75"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(75(a,1),11(a,2)),rewrite(75(8))].">
      <j1 rule="para" parents="75 11"/>
      <j2 rule="rewrite" parents="75"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      x * (x @ y) != x * y
    ]]></literal>
    <literal><![CDATA[
      (z @ z) @ u = y
    ]]></literal>
    <justification jstring="[para(75(a,1),12(a,1)),flip(b)].">
      <j1 rule="para" parents="75 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(50,a,75,a),flip(a)].">
      <j1 rule="hyper" parents="50 75"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(50,a,44,a)].">
      <j1 rule="hyper" parents="50 44"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(106)].">
      <j1 rule="new_symbol" parents="106"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(105),rewrite(108(1),108(3))].">
      <j1 rule="back_rewrite" parents="105"/>
      <j2 rule="rewrite" parents="108 108"/>
    </justification>
  </clause>

  <clause id="111">
    <literal><![CDATA[
      x * (x @ y) != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(104),rewrite(108(5),110(6))].">
      <j1 rule="back_rewrite" parents="104"/>
      <j2 rule="rewrite" parents="108 110"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(97),rewrite(108(3),110(4),110(4))].">
      <j1 rule="back_rewrite" parents="97"/>
      <j2 rule="rewrite" parents="108 110 110"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_rewrite(96),rewrite(108(1),110(2),110(2))].">
      <j1 rule="back_rewrite" parents="96"/>
      <j2 rule="rewrite" parents="108 110 110"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_rewrite(88),rewrite(108(1),110(2),108(4),110(5))].">
      <j1 rule="back_rewrite" parents="88"/>
      <j2 rule="rewrite" parents="108 110 108 110"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(76),rewrite(108(1),110(2),110(2))].">
      <j1 rule="back_rewrite" parents="76"/>
      <j2 rule="rewrite" parents="108 110 110"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(72),rewrite(108(1),110(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="72"/>
      <j2 rule="rewrite" parents="108 110"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_rewrite(119),rewrite(121(5))].">
      <j1 rule="back_rewrite" parents="119"/>
      <j2 rule="rewrite" parents="121"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(114,a,125,a),flip(a)].">
      <j1 rule="hyper" parents="114 125"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      x * y * (y @ y * x) = y * x
    ]]></literal>
    <justification jstring="[hyper(1,a,15,a)].">
      <j1 rule="hyper" parents="1 15"/>
    </justification>
  </clause>

  <clause id="170">
    <literal><![CDATA[
      A * B * C * B * A * (B @ C * A) != B * A * B * C * A
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[para(15(a,1),18(a,2,2,2))].">
      <j1 rule="para" parents="15 18"/>
    </justification>
  </clause>

  <clause id="214">
    <literal><![CDATA[
      x @ x * y != c_0
    ]]></literal>
    <literal><![CDATA[
      y @ x * y = c_0
    ]]></literal>
    <justification jstring="[para(123(a,1),114(a,1)),flip(a),flip(b)].">
      <j1 rule="para" parents="123 114"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(11,a,163,a),flip(a)].">
      <j1 rule="hyper" parents="11 163"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="232">
    <literal><![CDATA[
      (x @ y * x) * (y @ y * x) = c_0
    ]]></literal>
    <justification jstring="[para(123(a,1),163(a,1,2,2,2)),rewrite(142(8),121(6),123(10))].">
      <j1 rule="para" parents="123 163"/>
      <j2 rule="rewrite" parents="142 121 123"/>
    </justification>
  </clause>

  <clause id="233">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(1,a,220,a)].">
      <j1 rule="hyper" parents="1 220"/>
    </justification>
  </clause>

  <clause id="239">
    <literal><![CDATA[
      x * (x @ y) != x
    ]]></literal>
    <literal><![CDATA[
      x @ y = c_0
    ]]></literal>
    <justification jstring="[para(220(a,1),114(a,1)),rewrite(233(6)),flip(b)].">
      <j1 rule="para" parents="220 114"/>
      <j2 rule="rewrite" parents="233"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="248">
    <literal><![CDATA[
      (x @ y * x) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(232),rewrite(233(4))].">
      <j1 rule="back_rewrite" parents="232"/>
      <j2 rule="rewrite" parents="233"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      x @ y != c_0
    ]]></literal>
    <literal><![CDATA[
      y @ x * y = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(214),rewrite(233(2))].">
      <j1 rule="back_rewrite" parents="214"/>
      <j2 rule="rewrite" parents="233"/>
    </justification>
  </clause>

  <clause id="262">
    <literal><![CDATA[
      (x @ y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(123),rewrite(233(2))].">
      <j1 rule="back_rewrite" parents="123"/>
      <j2 rule="rewrite" parents="233"/>
    </justification>
  </clause>

  <clause id="265">
    <literal><![CDATA[
      (x @ y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[back_rewrite(54),rewrite(233(2))].">
      <j1 rule="back_rewrite" parents="54"/>
      <j2 rule="rewrite" parents="233"/>
    </justification>
  </clause>

  <clause id="276">
    <literal><![CDATA[
      (x @ y) * ((x @ y) @ y) = x @ y
    ]]></literal>
    <justification jstring="[para(108(a,1),17(a,1,1)),rewrite(115(6),108(6),121(7))].">
      <j1 rule="para" parents="108 17"/>
      <j2 rule="rewrite" parents="115 108 121"/>
    </justification>
  </clause>

  <clause id="286">
    <literal><![CDATA[
      x @ y * (y @ x) = x @ y * x
    ]]></literal>
    <justification jstring="[para(4(a,1),233(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 233"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="298">
    <literal><![CDATA[
      (x @ y * x) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(248(a,1),3(a,1,1)),rewrite(115(2)),flip(a)].">
      <j1 rule="para" parents="248 3"/>
      <j2 rule="rewrite" parents="115"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="317">
    <literal><![CDATA[
      (x @ y) @ x * y = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),262(a,1,2,2)),rewrite(286(3),6(6),233(4),298(8))].">
      <j1 rule="para" parents="4 262"/>
      <j2 rule="rewrite" parents="286 6 233 298"/>
    </justification>
  </clause>

  <clause id="318">
    <literal><![CDATA[
      x * y * (y @ x * y) = y * x
    ]]></literal>
    <justification jstring="[para(262(a,1),14(a,1,2,2)),rewrite(121(2)),flip(a)].">
      <j1 rule="para" parents="262 14"/>
      <j2 rule="rewrite" parents="121"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="402">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[hyper(22,a,17,a)].">
      <j1 rule="hyper" parents="22 17"/>
    </justification>
  </clause>

  <clause id="486">
    <literal><![CDATA[
      (x @ y) @ y = c_0
    ]]></literal>
    <justification jstring="[hyper(239,a,276,a)].">
      <j1 rule="hyper" parents="239 276"/>
    </justification>
  </clause>

  <clause id="487">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(276(a,1),4(a,1,2)),flip(a)].">
      <j1 rule="para" parents="276 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="494">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(253,a,486,a),rewrite(487(2),233(3))].">
      <j1 rule="hyper" parents="253 486"/>
      <j2 rule="rewrite" parents="487 233"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      x * (y @ x) != x
    ]]></literal>
    <literal><![CDATA[
      y @ x = c_0
    ]]></literal>
    <justification jstring="[para(494(a,1),111(a,1,2)),rewrite(121(2)),flip(a),flip(b)].">
      <j1 rule="para" parents="494 111"/>
      <j2 rule="rewrite" parents="121"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="688">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[hyper(22,a,318,a),flip(a)].">
      <j1 rule="hyper" parents="22 318"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="698">
    <literal><![CDATA[
      (x @ y) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(318(a,1),248(a,1,1,2)),rewrite(688(2),6(4),233(2),317(4),121(3),688(3),286(4),688(3))].">
      <j1 rule="para" parents="318 248"/>
      <j2 rule="rewrite" parents="688 6 233 317 121 688 286 688"/>
    </justification>
  </clause>

  <clause id="706">
    <literal><![CDATA[
      (x @ y) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(318(a,1),265(a,1,2,1,2)),rewrite(688(2),286(3),688(2),688(3),6(5),233(3),317(5),121(4))].">
      <j1 rule="para" parents="318 265"/>
      <j2 rule="rewrite" parents="688 286 688 688 6 233 317 121"/>
    </justification>
  </clause>

  <clause id="708">
    <literal><![CDATA[
      x * ((y @ z) @ x) = (y @ z) * x * (z @ y)
    ]]></literal>
    <justification jstring="[para(265(a,1),318(a,1,2,2,2)),rewrite(688(3),688(5),6(6),108(6),121(7),3(6),706(7),688(5),3(7))].">
      <j1 rule="para" parents="265 318"/>
      <j2 rule="rewrite" parents="688 688 6 108 121 3 706 688 3"/>
    </justification>
  </clause>

  <clause id="786">
    <literal><![CDATA[
      x @ y * z * x = x @ y * z
    ]]></literal>
    <justification jstring="[para(3(a,1),688(a,1,2))].">
      <j1 rule="para" parents="3 688"/>
    </justification>
  </clause>

  <clause id="999">
    <literal><![CDATA[
      x * (y @ z) * ((y @ x) @ (z @ x)) = (y @ z) * x
    ]]></literal>
    <justification jstring="[para(402(a,2),4(a,1,2,2))].">
      <j1 rule="para" parents="402 4"/>
    </justification>
  </clause>

  <clause id="1020">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = c_0
    ]]></literal>
    <justification jstring="[para(494(a,1),402(a,1,2)),rewrite(142(4)),flip(a)].">
      <j1 rule="para" parents="494 402"/>
      <j2 rule="rewrite" parents="142"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1024">
    <literal><![CDATA[
      (x @ y) * z * (y @ x) = z
    ]]></literal>
    <justification jstring="[para(402(a,2),487(a,1,1)),rewrite(1020(3),115(2),708(3)),flip(a)].">
      <j1 rule="para" parents="402 487"/>
      <j2 rule="rewrite" parents="1020 115 708"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1035">
    <literal><![CDATA[
      (x @ y) @ z = c_0
    ]]></literal>
    <justification jstring="[para(402(a,2),522(a,1,2)),rewrite(1020(3),121(2)),xx(a)].">
      <j1 rule="para" parents="402 522"/>
      <j2 rule="rewrite" parents="1020 121"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1038">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(402(a,2),698(a,1,1)),rewrite(1020(3),115(4))].">
      <j1 rule="para" parents="402 698"/>
      <j2 rule="rewrite" parents="1020 115"/>
    </justification>
  </clause>

  <clause id="1048">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(999),rewrite(1038(4),121(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="999"/>
      <j2 rule="rewrite" parents="1038 121"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1117">
    <literal><![CDATA[
      (x @ y * z) * u * (y @ x) * (z @ x) = u
    ]]></literal>
    <justification jstring="[para(6(a,1),1024(a,1,2,2))].">
      <j1 rule="para" parents="6 1024"/>
    </justification>
  </clause>

  <clause id="1240">
    <literal><![CDATA[
      x @ y * z = x @ z * y
    ]]></literal>
    <justification jstring="[para(14(a,1),786(a,1,2)),rewrite(6(5),1035(4),786(4),115(4),6(6),1035(5),115(6))].">
      <j1 rule="para" parents="14 786"/>
      <j2 rule="rewrite" parents="6 1035 786 115 6 1035 115"/>
    </justification>
  </clause>

  <clause id="1263">
    <literal><![CDATA[
      A * B * C * B * A * (B @ A * C) != B * A * B * C * A
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[back_rewrite(170),rewrite(1240(10))].">
      <j1 rule="back_rewrite" parents="170"/>
      <j2 rule="rewrite" parents="1240"/>
    </justification>
  </clause>

  <clause id="1331">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[ur(41,b,1263,a(flip)),rewrite(6(24),6(25),108(21),6(24),1048(26,R),115(27),3(27),3(26),3(25),3(24),3(23),1117(22),4(10),3(18),3(17),3(16)),xx(a)].">
      <j1 rule="ur" parents="41 1263"/>
      <j2 rule="rewrite" parents="6 6 108 6 1048 115 3 3 3 3 3 1117 4 3 3 3"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
