<?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>ED-gt.out</source>

<heading><![CDATA[
Prover9 (32) version 22-May-2007, May 2007.
Process 27451 was started by mccune on cleo,
Tue May 22 14:45:59 2007
The command was "/home/mccune/bin/prover9 -f gt.in ED.in".
]]></heading>

<proof number="1" length="119" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.73 (+ 0.01) seconds: D.
% Length of proof is 119.
% Level of proof is 28.
% Maximum clause weight is 23.
% Given clauses 243.
]]></comments>

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

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

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x' * x = e
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="5">
    <literal><![CDATA[
      x' * y' * x * y = x @ y
    ]]></literal>
    <justification jstring="[copy(4),flip(a)].">
      <j1 rule="copy" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x * y * z * y * x = y * x * z * x * y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

  <clause id="8">
    <literal><![CDATA[
      x' * x * y = y
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1,1)),rewrite([2(2)]),flip(a)].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x' * y' * x * y * z = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(5(a,1),1(a,1,1)),rewrite([1(7),1(6)]),flip(a)].">
      <j1 rule="para" parents="5 1"/>
      <j2 rule="rewrite" parents="1 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      e @ x = e
    ]]></literal>
    <justification jstring="[para(2(a,1),5(a,1,2,2)),rewrite([3(4),3(4)]),flip(a)].">
      <j1 rule="para" parents="2 5"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x' @ x = e
    ]]></literal>
    <justification jstring="[para(3(a,1),5(a,1,2,2)),rewrite([8(6)]),flip(a)].">
      <j1 rule="para" parents="3 5"/>
      <j2 rule="rewrite" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      x * y * y * x = y * x * x * y
    ]]></literal>
    <justification jstring="[para(2(a,1),6(a,1,2,2)),rewrite([2(6)])].">
      <j1 rule="para" parents="2 6"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * y * (x * y)' * y * x = y * x * e
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="3 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x'' * e = x
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,2))].">
      <j1 rule="para" parents="3 8"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x'' * (x * y)' * y = x' @ x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),5(a,1,2,2))].">
      <j1 rule="para" parents="8 5"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x'' * (x @ y) = y' * x * y
    ]]></literal>
    <justification jstring="[para(5(a,1),8(a,1,2))].">
      <j1 rule="para" parents="5 8"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x' * y * x * z * x * y = y * z * y * x
    ]]></literal>
    <justification jstring="[para(6(a,1),8(a,1,2))].">
      <j1 rule="para" parents="6 8"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x'' * y = x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),8(a,1,2))].">
      <j1 rule="para" parents="8 8"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      x' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_rewrite(27),rewrite([32(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="27"/>
      <j2 rule="rewrite" parents="32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      x * (x * y)' * y = x' @ x * y
    ]]></literal>
    <justification jstring="[back_rewrite(25),rewrite([32(6)])].">
      <j1 rule="back_rewrite" parents="25"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_rewrite(24),rewrite([32(4)])].">
      <j1 rule="back_rewrite" parents="24"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x * y * (x * y)' * y * x = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(20),rewrite([35(8)])].">
      <j1 rule="back_rewrite" parents="20"/>
      <j2 rule="rewrite" parents="35"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      e' = e
    ]]></literal>
    <justification jstring="[para(11(a,1),5(a,2)),rewrite([2(5),3(4),35(4)])].">
      <j1 rule="para" parents="11 5"/>
      <j2 rule="rewrite" parents="2 3 35"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x @ e = e
    ]]></literal>
    <justification jstring="[para(39(a,1),5(a,1,2,1)),rewrite([35(4),2(3),3(2)]),flip(a)].">
      <j1 rule="para" parents="39 5"/>
      <j2 rule="rewrite" parents="35 2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      (x' @ y) * x = x * (y @ x)
    ]]></literal>
    <justification jstring="[para(5(a,1),9(a,1,2)),rewrite([32(4)]),flip(a)].">
      <j1 rule="para" parents="5 9"/>
      <j2 rule="rewrite" parents="32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(6(a,1),9(a,1,2,2)),rewrite([8(7),8(5)]),flip(a)].">
      <j1 rule="para" parents="6 9"/>
      <j2 rule="rewrite" parents="8 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      x' * y * x * z = y * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2)),rewrite([32(5)]),flip(a)].">
      <j1 rule="para" parents="9 8"/>
      <j2 rule="rewrite" parents="32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="55">
    <literal><![CDATA[
      x' * y * x * z = (x @ y') * y * z
    ]]></literal>
    <justification jstring="[para(8(a,1),9(a,1,2,2,2)),rewrite([32(5)])].">
      <j1 rule="para" parents="8 9"/>
      <j2 rule="rewrite" parents="32"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      x * x' = e
    ]]></literal>
    <justification jstring="[para(32(a,1),3(a,1))].">
      <j1 rule="para" parents="32 3"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      x' * y * x * y' = x @ y'
    ]]></literal>
    <justification jstring="[para(32(a,1),5(a,1,2))].">
      <j1 rule="para" parents="32 5"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      x * y' * x' * y = x' @ y
    ]]></literal>
    <justification jstring="[para(32(a,1),5(a,1))].">
      <j1 rule="para" parents="32 5"/>
    </justification>
  </clause>

  <clause id="63">
    <literal><![CDATA[
      x * x' * y = y
    ]]></literal>
    <justification jstring="[para(32(a,1),8(a,1))].">
      <j1 rule="para" parents="32 8"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(32(a,1),35(a,1)),rewrite([35(2)]),flip(a)].">
      <j1 rule="para" parents="32 35"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (x' @ y) * z = x * y' * x' * y * z
    ]]></literal>
    <justification jstring="[para(32(a,1),9(a,1)),flip(a)].">
      <j1 rule="para" parents="32 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      x @ x' = e
    ]]></literal>
    <justification jstring="[para(64(a,1),12(a,1,1))].">
      <j1 rule="para" parents="64 12"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (x * y')' * y' * x = x * y' @ y
    ]]></literal>
    <justification jstring="[para(3(a,1),10(a,1,2,2,2)),rewrite([35(6)])].">
      <j1 rule="para" parents="3 10"/>
      <j2 rule="rewrite" parents="35"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      (x * y)' * y * x = x * y @ x
    ]]></literal>
    <justification jstring="[para(8(a,1),10(a,1,2))].">
      <j1 rule="para" parents="8 10"/>
    </justification>
  </clause>

  <clause id="77">
    <literal><![CDATA[
      x' * y' * x * y * z @ u = (x @ y) * z @ u
    ]]></literal>
    <justification jstring="[para(9(a,2),10(a,2,1)),rewrite([10(9)]),flip(a)].">
      <j1 rule="para" parents="9 10"/>
      <j2 rule="rewrite" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      x * y' @ y = x * y' @ x
    ]]></literal>
    <justification jstring="[back_rewrite(69),rewrite([74(6)]),flip(a)].">
      <j1 rule="back_rewrite" parents="69"/>
      <j2 rule="rewrite" parents="74"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(37),rewrite([74(4)])].">
      <j1 rule="back_rewrite" parents="37"/>
      <j2 rule="rewrite" parents="74"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x * y * (x * y)' = e
    ]]></literal>
    <justification jstring="[para(58(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="58 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x * y @ y' = x * y @ x
    ]]></literal>
    <justification jstring="[para(58(a,1),10(a,1,2,2,2)),rewrite([64(4),35(4),74(4)]),flip(a)].">
      <j1 rule="para" parents="58 10"/>
      <j2 rule="rewrite" parents="64 35 74"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      x * (y * x)' = y'
    ]]></literal>
    <justification jstring="[para(82(a,1),8(a,1,2)),rewrite([35(3)]),flip(a)].">
      <j1 rule="para" parents="82 8"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="118">
    <literal><![CDATA[
      (x @ y) * z * (x' * y' * x * y * z)' = e
    ]]></literal>
    <justification jstring="[para(9(a,2),82(a,1,2,2,1))].">
      <j1 rule="para" parents="9 82"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      (x * y)' = y' * x'
    ]]></literal>
    <justification jstring="[para(116(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="116 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      (x @ y)' = y @ x
    ]]></literal>
    <justification jstring="[para(9(a,2),116(a,1,2,1)),rewrite([129(7),129(5),129(3),129(2),1(5),64(7),1(6),1(5),64(8),1(7),1(6),1(5),5(6),63(4)]),flip(a)].">
      <j1 rule="para" parents="9 116"/>
      <j2 rule="rewrite" parents="129 129 129 129 1 64 1 1 64 1 1 1 5 63"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_rewrite(118),rewrite([129(8),129(6),129(4),129(3),1(6),64(8),1(7),1(6),64(9),1(8),1(7),1(6),5(7),63(5)])].">
      <j1 rule="back_rewrite" parents="118"/>
      <j2 rule="rewrite" parents="129 129 129 129 1 64 1 1 64 1 1 1 5 63"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(74),rewrite([129(2),1(5),5(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="74"/>
      <j2 rule="rewrite" parents="129 1 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="152">
    <literal><![CDATA[
      x' @ x * y = x' @ y
    ]]></literal>
    <justification jstring="[back_rewrite(34),rewrite([129(2),1(4),61(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="34"/>
      <j2 rule="rewrite" parents="129 1 61"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="158">
    <literal><![CDATA[
      x * y @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(85),rewrite([145(5)])].">
      <j1 rule="back_rewrite" parents="85"/>
      <j2 rule="rewrite" parents="145"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(81),rewrite([145(2)])].">
      <j1 rule="back_rewrite" parents="81"/>
      <j2 rule="rewrite" parents="145"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x * y' @ y = y' @ x
    ]]></literal>
    <justification jstring="[back_rewrite(80),rewrite([145(6)])].">
      <j1 rule="back_rewrite" parents="80"/>
      <j2 rule="rewrite" parents="145"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      (x @ y) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(130(a,1),8(a,1,1))].">
      <j1 rule="para" parents="130 8"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      x' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(63(a,1),145(a,1,1)),flip(a)].">
      <j1 rule="para" parents="63 145"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[para(145(a,1),130(a,1,1)),rewrite([130(2)]),flip(a)].">
      <j1 rule="para" parents="145 130"/>
      <j2 rule="rewrite" parents="130"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="188">
    <literal><![CDATA[
      x' @ y' * x * y = x' @ (x @ y)
    ]]></literal>
    <justification jstring="[para(5(a,1),185(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5 185"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="192">
    <literal><![CDATA[
      x @ x' * y = x @ y
    ]]></literal>
    <justification jstring="[para(63(a,1),185(a,1,2)),flip(a)].">
      <j1 rule="para" parents="63 185"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      x' @ y * x = y @ x
    ]]></literal>
    <justification jstring="[para(158(a,1),130(a,1,1)),rewrite([130(2)]),flip(a)].">
      <j1 rule="para" parents="158 130"/>
      <j2 rule="rewrite" parents="130"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="237">
    <literal><![CDATA[
      x @ y * x' = y @ x'
    ]]></literal>
    <justification jstring="[para(64(a,1),218(a,1,1))].">
      <j1 rule="para" parents="64 218"/>
    </justification>
  </clause>

  <clause id="246">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(18(a,1),9(a,1,2,2)),rewrite([8(6),8(4)]),flip(a)].">
      <j1 rule="para" parents="18 9"/>
      <j2 rule="rewrite" parents="8 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="269">
    <literal><![CDATA[
      (x @ y) * z = z * x' * y' * x * y * ((x @ y) @ z)
    ]]></literal>
    <justification jstring="[para(9(a,2),159(a,1,2)),flip(a)].">
      <j1 rule="para" parents="9 159"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="287">
    <literal><![CDATA[
      x' * (y @ z) * x * u = (x @ (z @ y)) * (y @ z) * u
    ]]></literal>
    <justification jstring="[para(167(a,1),9(a,1,2,2,2)),rewrite([130(3)])].">
      <j1 rule="para" parents="167 9"/>
      <j2 rule="rewrite" parents="130"/>
    </justification>
  </clause>

  <clause id="362">
    <literal><![CDATA[
      (x @ y) @ y * x = e
    ]]></literal>
    <justification jstring="[para(246(a,1),5(a,1,2,2)),rewrite([130(2),129(3),1(6),5(6),139(3)]),flip(a)].">
      <j1 rule="para" parents="246 5"/>
      <j2 rule="rewrite" parents="130 129 1 5 139"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="364">
    <literal><![CDATA[
      (x' @ (x @ y)) * (y @ x') = x @ y
    ]]></literal>
    <justification jstring="[para(5(a,1),246(a,2)),rewrite([188(5),1(8),1(7),60(8)])].">
      <j1 rule="para" parents="5 246"/>
      <j2 rule="rewrite" parents="188 1 1 60"/>
    </justification>
  </clause>

  <clause id="401">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(63(a,1),362(a,1,2)),rewrite([183(3)])].">
      <j1 rule="para" parents="63 362"/>
      <j2 rule="rewrite" parents="183"/>
    </justification>
  </clause>

  <clause id="420">
    <literal><![CDATA[
      x @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(401(a,1),130(a,1,1)),rewrite([39(2)]),flip(a)].">
      <j1 rule="para" parents="401 130"/>
      <j2 rule="rewrite" parents="39"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="421">
    <literal><![CDATA[
      (x @ y) @ y' * x = e
    ]]></literal>
    <justification jstring="[para(183(a,1),401(a,1,1))].">
      <j1 rule="para" parents="183 401"/>
    </justification>
  </clause>

  <clause id="422">
    <literal><![CDATA[
      (x @ y) @ y' = e
    ]]></literal>
    <justification jstring="[para(218(a,1),401(a,1,1))].">
      <j1 rule="para" parents="218 401"/>
    </justification>
  </clause>

  <clause id="423">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[para(401(a,1),159(a,1,2,2)),rewrite([35(3)]),flip(a)].">
      <j1 rule="para" parents="401 159"/>
      <j2 rule="rewrite" parents="35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="425">
    <literal><![CDATA[
      x @ (x' @ y) = e
    ]]></literal>
    <justification jstring="[para(401(a,1),237(a,2)),rewrite([423(4),192(5)])].">
      <j1 rule="para" parents="401 237"/>
      <j2 rule="rewrite" parents="423 192"/>
    </justification>
  </clause>

  <clause id="442">
    <literal><![CDATA[
      (x' @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(420(a,1),160(a,2)),rewrite([423(4),183(5)])].">
      <j1 rule="para" parents="420 160"/>
      <j2 rule="rewrite" parents="423 183"/>
    </justification>
  </clause>

  <clause id="478">
    <literal><![CDATA[
      x' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(64(a,1),425(a,1,2,1))].">
      <j1 rule="para" parents="64 425"/>
    </justification>
  </clause>

  <clause id="482">
    <literal><![CDATA[
      x @ (y @ x) = e
    ]]></literal>
    <justification jstring="[para(218(a,1),425(a,1,2))].">
      <j1 rule="para" parents="218 425"/>
    </justification>
  </clause>

  <clause id="484">
    <literal><![CDATA[
      x @ y' = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(364),rewrite([478(3),2(4)])].">
      <j1 rule="back_rewrite" parents="364"/>
      <j2 rule="rewrite" parents="478 2"/>
    </justification>
  </clause>

  <clause id="492">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(482(a,1),159(a,1,2,2)),rewrite([35(3)])].">
      <j1 rule="para" parents="482 159"/>
      <j2 rule="rewrite" parents="35"/>
    </justification>
  </clause>

  <clause id="579">
    <literal><![CDATA[
      x' @ y = y @ x
    ]]></literal>
    <justification jstring="[para(442(a,1),33(a,2,2)),rewrite([49(4),8(4),35(5)]),flip(a)].">
      <j1 rule="para" parents="442 33"/>
      <j2 rule="rewrite" parents="49 8 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="591">
    <literal><![CDATA[
      (C @ A') * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(484(a,2),7(a,1,1))].">
      <j1 rule="para" parents="484 7"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(130(a,1),484(a,1,2)),flip(a)].">
      <j1 rule="para" parents="130 484"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="601">
    <literal><![CDATA[
      x' @ y' = x @ y
    ]]></literal>
    <justification jstring="[para(484(a,2),145(a,1)),rewrite([129(2),237(4)])].">
      <j1 rule="para" parents="484 145"/>
      <j2 rule="rewrite" parents="129 237"/>
    </justification>
  </clause>

  <clause id="609">
    <literal><![CDATA[
      x' @ y = x @ y'
    ]]></literal>
    <justification jstring="[para(484(a,2),160(a,1)),rewrite([129(3),64(2),185(3)]),flip(a)].">
      <j1 rule="para" parents="484 160"/>
      <j2 rule="rewrite" parents="129 64 185"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="669">
    <literal><![CDATA[
      C' * A * C * A' * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(9(a,2),591(a,1)),rewrite([64(5)])].">
      <j1 rule="para" parents="9 591"/>
      <j2 rule="rewrite" parents="64"/>
    </justification>
  </clause>

  <clause id="992">
    <literal><![CDATA[
      x' * (y @ z) * x * z * y = y * z
    ]]></literal>
    <justification jstring="[para(50(a,2),8(a,1,2))].">
      <j1 rule="para" parents="50 8"/>
    </justification>
  </clause>

  <clause id="1050">
    <literal><![CDATA[
      (x @ y) * z * y * (y @ x) = z * y
    ]]></literal>
    <justification jstring="[para(33(a,1),50(a,1,2,2)),rewrite([158(3),1(8),58(7),35(7)])].">
      <j1 rule="para" parents="33 50"/>
      <j2 rule="rewrite" parents="158 1 58 35"/>
    </justification>
  </clause>

  <clause id="1088">
    <literal><![CDATA[
      C' * (C @ A' * (B @ C)) * C * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(50(a,2),669(a,1,2)),rewrite([1(19),492(18),63(20)])].">
      <j1 rule="para" parents="50 669"/>
      <j2 rule="rewrite" parents="1 492 63"/>
    </justification>
  </clause>

  <clause id="1213">
    <literal><![CDATA[
      x * (x @ y') * z = y * x * y' * z
    ]]></literal>
    <justification jstring="[para(64(a,1),53(a,1,1)),flip(a)].">
      <j1 rule="para" parents="64 53"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1252">
    <literal><![CDATA[
      x * (y @ x) * x' * z = (y @ x) * z
    ]]></literal>
    <justification jstring="[para(422(a,1),53(a,2,2,1)),rewrite([64(2),2(8)])].">
      <j1 rule="para" parents="422 53"/>
      <j2 rule="rewrite" parents="64 2"/>
    </justification>
  </clause>

  <clause id="1269">
    <literal><![CDATA[
      (x @ y) * y' * (y @ x) * z = y' * z
    ]]></literal>
    <justification jstring="[para(478(a,1),53(a,2,2,1)),rewrite([130(2),2(9)])].">
      <j1 rule="para" parents="478 53"/>
      <j2 rule="rewrite" parents="130 2"/>
    </justification>
  </clause>

  <clause id="1298">
    <literal><![CDATA[
      x' * (x @ y) * x * z = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(421(a,1),53(a,2,2,1)),rewrite([129(3),64(3),1(6),1(8),1252(7),2(8)])].">
      <j1 rule="para" parents="421 53"/>
      <j2 rule="rewrite" parents="129 64 1 1 1252 2"/>
    </justification>
  </clause>

  <clause id="1338">
    <literal><![CDATA[
      (C @ A' * (B @ C)) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_rewrite(1088),rewrite([1298(17)])].">
      <j1 rule="back_rewrite" parents="1088"/>
      <j2 rule="rewrite" parents="1298"/>
    </justification>
  </clause>

  <clause id="1344">
    <literal><![CDATA[
      ((C @ B) * A @ C) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(579(a,2),1338(a,1,1)),rewrite([129(7),130(4),64(6)])].">
      <j1 rule="para" parents="579 1338"/>
      <j2 rule="rewrite" parents="129 130 64"/>
    </justification>
  </clause>

  <clause id="1350">
    <literal><![CDATA[
      (C' @ (C @ B) * A) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(579(a,2),1344(a,1,1))].">
      <j1 rule="para" parents="579 1344"/>
    </justification>
  </clause>

  <clause id="1420">
    <literal><![CDATA[
      ((x @ y) @ z') * z * u = (y @ x) * z * (x @ y) * u
    ]]></literal>
    <justification jstring="[para(130(a,1),55(a,1,1)),flip(a)].">
      <j1 rule="para" parents="130 55"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1505">
    <literal><![CDATA[
      C * A' * C' * A * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(9(a,2),1350(a,1)),rewrite([64(3),129(7),130(7),1(18),1(20),1269(19)])].">
      <j1 rule="para" parents="9 1350"/>
      <j2 rule="rewrite" parents="64 129 130 1 1 1269"/>
    </justification>
  </clause>

  <clause id="1866">
    <literal><![CDATA[
      x' * y * z * x * z' * y' = x @ z' * y'
    ]]></literal>
    <justification jstring="[para(1(a,1),60(a,1,2)),rewrite([129(3),129(10)])].">
      <j1 rule="para" parents="1 60"/>
      <j2 rule="rewrite" parents="129 129"/>
    </justification>
  </clause>

  <clause id="1943">
    <literal><![CDATA[
      (x @ (y @ z)) * y = y
    ]]></literal>
    <justification jstring="[para(423(a,1),9(a,1,2,2,2)),rewrite([130(3),1050(6),8(3)]),flip(a)].">
      <j1 rule="para" parents="423 9"/>
      <j2 rule="rewrite" parents="130 1050 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1957">
    <literal><![CDATA[
      x @ ((x @ y) @ z) = (x @ y) @ z
    ]]></literal>
    <justification jstring="[para(1943(a,1),5(a,1,2,2)),rewrite([130(3),3(4),35(4),598(5)]),flip(a)].">
      <j1 rule="para" parents="1943 5"/>
      <j2 rule="rewrite" parents="130 3 35 598"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1966">
    <literal><![CDATA[
      (x @ y) @ z = e
    ]]></literal>
    <justification jstring="[para(1943(a,1),218(a,1,2)),rewrite([609(2),67(2),598(4),1957(4)]),flip(a)].">
      <j1 rule="para" parents="1943 218"/>
      <j2 rule="rewrite" parents="609 67 598 1957"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1977">
    <literal><![CDATA[
      x @ (y @ z) = e
    ]]></literal>
    <justification jstring="[para(1943(a,1),33(a,1,2)),rewrite([3(2),598(6),1966(5),40(5),35(5)]),flip(a)].">
      <j1 rule="para" parents="1943 33"/>
      <j2 rule="rewrite" parents="3 598 1966 40 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1991">
    <literal><![CDATA[
      (x @ y) * z * (y @ x) * u = z * u
    ]]></literal>
    <justification jstring="[back_rewrite(1420),rewrite([1966(3),2(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="1420"/>
      <j2 rule="rewrite" parents="1966 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1999">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(269),rewrite([1966(6),35(6),5(7)])].">
      <j1 rule="back_rewrite" parents="269"/>
      <j2 rule="rewrite" parents="1966 35 5"/>
    </justification>
  </clause>

  <clause id="2028">
    <literal><![CDATA[
      x' * (y @ z) * x * u = (y @ z) * u
    ]]></literal>
    <justification jstring="[back_rewrite(287),rewrite([1977(7),2(9)])].">
      <j1 rule="back_rewrite" parents="287"/>
      <j2 rule="rewrite" parents="1977 2"/>
    </justification>
  </clause>

  <clause id="2102">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[back_rewrite(992),rewrite([2028(6)])].">
      <j1 rule="back_rewrite" parents="992"/>
      <j2 rule="rewrite" parents="2028"/>
    </justification>
  </clause>

  <clause id="2314">
    <literal><![CDATA[
      x * (y @ z) * x' * u = (y @ z) * u
    ]]></literal>
    <justification jstring="[para(167(a,1),66(a,2,2,2,2)),rewrite([1977(3),2(4),130(4)]),flip(a)].">
      <j1 rule="para" parents="167 66"/>
      <j2 rule="rewrite" parents="1977 2 130"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2670">
    <literal><![CDATA[
      x' * y' * x * y * z @ u = z @ u
    ]]></literal>
    <justification jstring="[para(77(a,2),5(a,2)),rewrite([129(3),130(3),1(7),1(9),1991(8),5(5)]),flip(a)].">
      <j1 rule="para" parents="77 5"/>
      <j2 rule="rewrite" parents="129 130 1 1 1991 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2684">
    <literal><![CDATA[
      (x @ y) * z @ u = z @ u
    ]]></literal>
    <justification jstring="[para(77(a,1),130(a,2)),rewrite([130(8),2670(7)]),flip(a)].">
      <j1 rule="para" parents="77 130"/>
      <j2 rule="rewrite" parents="130 2670"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2685">
    <literal><![CDATA[
      x @ (y @ z) * u = x @ u
    ]]></literal>
    <justification jstring="[para(77(a,2),130(a,1,1)),rewrite([2670(7),130(2)]),flip(a)].">
      <j1 rule="para" parents="77 130"/>
      <j2 rule="rewrite" parents="2670 130"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2692">
    <literal><![CDATA[
      x' * y' * z' * x * z * y * u @ v = u @ v
    ]]></literal>
    <justification jstring="[para(129(a,1),77(a,1,1,2,1)),rewrite([1(6),1(8),2684(14)])].">
      <j1 rule="para" parents="129 77"/>
      <j2 rule="rewrite" parents="1 1 2684"/>
    </justification>
  </clause>

  <clause id="2716">
    <literal><![CDATA[
      x * y @ z = y * x @ z
    ]]></literal>
    <justification jstring="[para(31(a,2),77(a,1,1,2,2)),rewrite([2692(11),2684(6)])].">
      <j1 rule="para" parents="31 77"/>
      <j2 rule="rewrite" parents="2692 2684"/>
    </justification>
  </clause>

  <clause id="2901">
    <literal><![CDATA[
      x @ y * z = x @ z * y
    ]]></literal>
    <justification jstring="[para(2102(a,1),2685(a,1,2))].">
      <j1 rule="para" parents="2102 2685"/>
    </justification>
  </clause>

  <clause id="2971">
    <literal><![CDATA[
      x * y @ z = z @ x' * y'
    ]]></literal>
    <justification jstring="[para(2716(a,1),484(a,2)),rewrite([129(2)]),flip(a)].">
      <j1 rule="para" parents="2716 484"/>
      <j2 rule="rewrite" parents="129"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3161">
    <literal><![CDATA[
      C * (A @ C) * (B @ C) * C' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(55(a,1),1505(a,1,2)),rewrite([64(5),1999(10,R)])].">
      <j1 rule="para" parents="55 1505"/>
      <j2 rule="rewrite" parents="64 1999"/>
    </justification>
  </clause>

  <clause id="3162">
    <literal><![CDATA[
      C * (A @ C) * B' * C' * B != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(9(a,2),3161(a,1,2,2)),rewrite([58(13),35(11)])].">
      <j1 rule="para" parents="9 3161"/>
      <j2 rule="rewrite" parents="58 35"/>
    </justification>
  </clause>

  <clause id="3295">
    <literal><![CDATA[
      C * (B @ C) * (A @ C) * C' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(50(a,2),3162(a,1,2)),rewrite([2901(8),152(8),601(6),1(14),58(13),35(11)])].">
      <j1 rule="para" parents="50 3162"/>
      <j2 rule="rewrite" parents="2901 152 601 1 58 35"/>
    </justification>
  </clause>

  <clause id="3471">
    <literal><![CDATA[
      B * C * B' * (A @ C) * C' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(484(a,2),3295(a,1,2,1)),rewrite([1213(13)])].">
      <j1 rule="para" parents="484 3295"/>
      <j2 rule="rewrite" parents="1213"/>
    </justification>
  </clause>

  <clause id="3634">
    <literal><![CDATA[
      B * (B @ C) * (A @ C) * B' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(50(a,2),3471(a,1,2)),rewrite([2685(10),601(6),1(14),2314(15)])].">
      <j1 rule="para" parents="50 3471"/>
      <j2 rule="rewrite" parents="2685 601 1 2314"/>
    </justification>
  </clause>

  <clause id="3826">
    <literal><![CDATA[
      C' * B * C * (A @ C) * B' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(9(a,2),3634(a,1,2)),rewrite([63(18)])].">
      <j1 rule="para" parents="9 3634"/>
      <j2 rule="rewrite" parents="63"/>
    </justification>
  </clause>

  <clause id="3834">
    <literal><![CDATA[
      C @ A' * B' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(484(a,2),3826(a,1,2,2,2,1)),rewrite([1213(12),1866(14)])].">
      <j1 rule="para" parents="484 3826"/>
      <j2 rule="rewrite" parents="1213 1866"/>
    </justification>
  </clause>

  <clause id="3835">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[resolve(3834,a,2971,a(flip))].">
      <j1 rule="resolve" parents="3834 2971"/>
    </justification>
  </clause>

</proof>

</proofs>
