<?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 August-2006A, August 2006.
Process 11355 was started by mccune on cleo.thornwood,
Sat Aug 12 20:59:56 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 0.13 (+ 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="1">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

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

  <clause id="5">
    <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">
    <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="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="16">
    <literal><![CDATA[
      x * y * z * (y @ x) * (z @ x) = y * z * x
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,2)),rewrite(6(2),3(8))].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="rewrite" parents="6 3"/>
    </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 != 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="25">
    <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="33">
    <literal><![CDATA[
      x * y * z != x * u * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ y) = z
    ]]></literal>
    <justification jstring="[para(4(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34">
    <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="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,25,a)].">
      <j1 rule="hyper" parents="8 25"/>
    </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="59">
    <literal><![CDATA[
      x * y * z != u * x * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ x * y) = z
    ]]></literal>
    <justification jstring="[para(3(a,1),11(a,1))].">
      <j1 rule="para" parents="3 11"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(22,a,51,a),flip(a)].">
      <j1 rule="hyper" parents="22 51"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      (u @ u) @ y = z
    ]]></literal>
    <justification jstring="[para(51(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="51 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      (x @ x) @ y * z = (y @ y * z) * (z @ y * z)
    ]]></literal>
    <justification jstring="[para(65(a,2),6(a,1))].">
      <j1 rule="para" parents="65 6"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(65(a,2),44(a,1,1))].">
      <j1 rule="para" parents="65 44"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(65(a,2),51(a,1,2,1))].">
      <j1 rule="para" parents="65 51"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(11,a,84,a)].">
      <j1 rule="hyper" parents="11 84"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      (z @ z) @ u = x
    ]]></literal>
    <justification jstring="[para(84(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="84 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="102">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(84(a,1),11(a,2)),rewrite(84(8))].">
      <j1 rule="para" parents="84 11"/>
      <j2 rule="rewrite" parents="84"/>
    </justification>
  </clause>

  <clause id="104">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(65(a,2),84(a,1,1,1))].">
      <j1 rule="para" parents="65 84"/>
    </justification>
  </clause>

  <clause id="109">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(50,a,84,a),flip(a)].">
      <j1 rule="hyper" parents="50 84"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(50,a,44,a)].">
      <j1 rule="hyper" parents="50 44"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(110)].">
      <j1 rule="new_symbol" parents="110"/>
    </justification>
  </clause>

  <clause id="114">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(109),rewrite(112(1),112(3))].">
      <j1 rule="back_rewrite" parents="109"/>
      <j2 rule="rewrite" parents="112 112"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_rewrite(104),rewrite(112(1),114(2),114(2))].">
      <j1 rule="back_rewrite" parents="104"/>
      <j2 rule="rewrite" parents="112 114 114"/>
    </justification>
  </clause>

  <clause id="119">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(102),rewrite(112(3),114(4),114(4))].">
      <j1 rule="back_rewrite" parents="102"/>
      <j2 rule="rewrite" parents="112 114 114"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(96),rewrite(112(3),114(4))].">
      <j1 rule="back_rewrite" parents="96"/>
      <j2 rule="rewrite" parents="112 114"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_rewrite(95),rewrite(112(1),114(2),112(4),114(5))].">
      <j1 rule="back_rewrite" parents="95"/>
      <j2 rule="rewrite" parents="112 114 112 114"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(85),rewrite(112(1),114(2),114(2))].">
      <j1 rule="back_rewrite" parents="85"/>
      <j2 rule="rewrite" parents="112 114 114"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(81),rewrite(112(1),114(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="81"/>
      <j2 rule="rewrite" parents="112 114"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = z
    ]]></literal>
    <justification jstring="[back_rewrite(74),rewrite(112(5),114(6))].">
      <j1 rule="back_rewrite" parents="74"/>
      <j2 rule="rewrite" parents="112 114"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_rewrite(123),rewrite(124(5))].">
      <j1 rule="back_rewrite" parents="123"/>
      <j2 rule="rewrite" parents="124"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(119,a,130,a),flip(a)].">
      <j1 rule="hyper" parents="119 130"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="165">
    <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="171">
    <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="205">
    <literal><![CDATA[
      (x @ y) * ((x @ y) @ y) = x @ y
    ]]></literal>
    <justification jstring="[para(112(a,1),17(a,1,1)),rewrite(118(6),112(6),124(7))].">
      <j1 rule="para" parents="112 17"/>
      <j2 rule="rewrite" parents="118 112 124"/>
    </justification>
  </clause>

  <clause id="217">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      x @ y = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),129(a,1)),flip(b)].">
      <j1 rule="para" parents="4 129"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(11,a,165,a),flip(a)].">
      <j1 rule="hyper" parents="11 165"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="265">
    <literal><![CDATA[
      (x @ y * x) * (y @ y * x) = c_0
    ]]></literal>
    <justification jstring="[para(126(a,1),165(a,1,2,2,2)),rewrite(163(8),124(6),126(10))].">
      <j1 rule="para" parents="126 165"/>
      <j2 rule="rewrite" parents="163 124 126"/>
    </justification>
  </clause>

  <clause id="268">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(205(a,1),4(a,1,2)),flip(a)].">
      <j1 rule="para" parents="205 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="287">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      z = y
    ]]></literal>
    <justification jstring="[para(118(a,1),33(a,1,2)),rewrite(124(3),163(5),124(5))].">
      <j1 rule="para" parents="118 33"/>
      <j2 rule="rewrite" parents="124 163 124"/>
    </justification>
  </clause>

  <clause id="292">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(217,a,268,a(flip))].">
      <j1 rule="hyper" parents="217 268"/>
    </justification>
  </clause>

  <clause id="304">
    <literal><![CDATA[
      x * (y @ x) != x
    ]]></literal>
    <literal><![CDATA[
      y @ x = c_0
    ]]></literal>
    <justification jstring="[para(268(a,1),122(a,1)),flip(b)].">
      <j1 rule="para" parents="268 122"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="317">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[hyper(34,a,17,a)].">
      <j1 rule="hyper" parents="34 17"/>
    </justification>
  </clause>

  <clause id="321">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = c_0
    ]]></literal>
    <justification jstring="[para(124(a,1),34(a,1,2))].">
      <j1 rule="para" parents="124 34"/>
    </justification>
  </clause>

  <clause id="332">
    <literal><![CDATA[
      (x @ y * x) @ (y @ y * x) = c_0
    ]]></literal>
    <justification jstring="[para(126(a,1),321(a,1)),rewrite(265(6)),xx(a)].">
      <j1 rule="para" parents="126 321"/>
      <j2 rule="rewrite" parents="265"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="402">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(287,a,253,a),flip(a)].">
      <j1 rule="hyper" parents="287 253"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="422">
    <literal><![CDATA[
      (x @ y * x) @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(332),rewrite(402(4))].">
      <j1 rule="back_rewrite" parents="332"/>
      <j2 rule="rewrite" parents="402"/>
    </justification>
  </clause>

  <clause id="424">
    <literal><![CDATA[
      (x @ y * x) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(265),rewrite(402(4))].">
      <j1 rule="back_rewrite" parents="265"/>
      <j2 rule="rewrite" parents="402"/>
    </justification>
  </clause>

  <clause id="442">
    <literal><![CDATA[
      (x @ y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[back_rewrite(54),rewrite(402(2))].">
      <j1 rule="back_rewrite" parents="54"/>
      <j2 rule="rewrite" parents="402"/>
    </justification>
  </clause>

  <clause id="448">
    <literal><![CDATA[
      x @ y * (y @ x) = x @ y * x
    ]]></literal>
    <justification jstring="[para(4(a,1),402(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 402"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="489">
    <literal><![CDATA[
      (x @ y * x) * z * (y @ x) = z * (z @ (y @ x))
    ]]></literal>
    <justification jstring="[para(422(a,1),16(a,1,2,2,2,1)),rewrite(118(7),442(8)),flip(a)].">
      <j1 rule="para" parents="422 16"/>
      <j2 rule="rewrite" parents="118 442"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      x * (y @ z) * ((y @ x) @ (z @ x)) = (y @ z) * x
    ]]></literal>
    <justification jstring="[para(317(a,2),4(a,1,2,2))].">
      <j1 rule="para" parents="317 4"/>
    </justification>
  </clause>

  <clause id="558">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = c_0
    ]]></literal>
    <justification jstring="[para(292(a,1),317(a,1,2)),rewrite(163(4)),flip(a)].">
      <j1 rule="para" parents="292 317"/>
      <j2 rule="rewrite" parents="163"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="563">
    <literal><![CDATA[
      (x @ y) @ z = c_0
    ]]></literal>
    <justification jstring="[para(317(a,2),304(a,1,2)),rewrite(558(3),124(2)),xx(a)].">
      <j1 rule="para" parents="317 304"/>
      <j2 rule="rewrite" parents="558 124"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="566">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(317(a,2),424(a,1,1)),rewrite(558(7),118(4))].">
      <j1 rule="para" parents="317 424"/>
      <j2 rule="rewrite" parents="558 118"/>
    </justification>
  </clause>

  <clause id="575">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(535),rewrite(566(4),124(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="535"/>
      <j2 rule="rewrite" parents="566 124"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="588">
    <literal><![CDATA[
      (x @ y * x) * z * (y @ x) = z
    ]]></literal>
    <justification jstring="[back_rewrite(489),rewrite(566(7),124(7))].">
      <j1 rule="back_rewrite" parents="489"/>
      <j2 rule="rewrite" parents="566 124"/>
    </justification>
  </clause>

  <clause id="601">
    <literal><![CDATA[
      x * (x @ y * x) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(59,a,14,a),rewrite(575(5))].">
      <j1 rule="hyper" parents="59 14"/>
      <j2 rule="rewrite" parents="575"/>
    </justification>
  </clause>

  <clause id="731">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[para(4(a,1),448(a,2,2)),rewrite(448(5),601(5),6(5),448(3),563(6),124(4),6(6),402(4),563(6),124(5))].">
      <j1 rule="para" parents="4 448"/>
      <j2 rule="rewrite" parents="448 601 6 448 563 124 6 402 563 124"/>
    </justification>
  </clause>

  <clause id="768">
    <literal><![CDATA[
      (x @ y) * z * (y @ x) = z
    ]]></literal>
    <justification jstring="[back_rewrite(588),rewrite(731(2))].">
      <j1 rule="back_rewrite" parents="588"/>
      <j2 rule="rewrite" parents="731"/>
    </justification>
  </clause>

  <clause id="789">
    <literal><![CDATA[
      x @ y * z * x = x @ y * z
    ]]></literal>
    <justification jstring="[para(3(a,1),731(a,1,2))].">
      <j1 rule="para" parents="3 731"/>
    </justification>
  </clause>

  <clause id="831">
    <literal><![CDATA[
      (x @ y * z) * u * (y @ x) * (z @ x) = u
    ]]></literal>
    <justification jstring="[para(6(a,1),768(a,1,2,2))].">
      <j1 rule="para" parents="6 768"/>
    </justification>
  </clause>

  <clause id="923">
    <literal><![CDATA[
      x @ y * z = x @ z * y
    ]]></literal>
    <justification jstring="[para(14(a,1),789(a,1,2)),rewrite(6(5),563(4),789(4),118(4),6(6),563(5),118(6))].">
      <j1 rule="para" parents="14 789"/>
      <j2 rule="rewrite" parents="6 563 789 118 6 563 118"/>
    </justification>
  </clause>

  <clause id="945">
    <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(171),rewrite(923(10))].">
      <j1 rule="back_rewrite" parents="171"/>
      <j2 rule="rewrite" parents="923"/>
    </justification>
  </clause>

  <clause id="990">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(E)
    ]]></attribute>
    <justification jstring="[ur(41,b,945,a(flip)),rewrite(6(24),6(25),112(21),6(24),575(26,R),118(27),3(27),3(26),3(25),3(24),3(23),831(22),4(10),3(18),3(17),3(16)),xx(a)].">
      <j1 rule="ur" parents="41 945"/>
      <j2 rule="rewrite" parents="6 6 112 6 575 118 3 3 3 3 3 831 4 3 3 3"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
