<?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 June-2006C, June 2006.
Process 13126 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:39 2006
The command was "/home/mccune/bin/prover9 -f gt.in ED.in".
]]></heading>

<proof number="1" length="89" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.69 (+ 0.01) seconds: D.
% Length of proof is 89.
% Level of proof is 22.
% Maximum clause weight is 23.
% Given clauses 194.
]]></comments>

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

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

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

  <clause id="9">
    <literal><![CDATA[
      x ' * x = e
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <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="11">
    <literal><![CDATA[
      x * y * z * y * x = y * x * z * x * y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

  <clause id="13">
    <literal><![CDATA[
      x ' * x * y = y
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1,1)),demod(8(2)),flip(a)].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="demod" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x ' * y ' * x * y * z = (x @ y) * z
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1,1)),demod(7(7),7(6)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="demod" parents="7 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      (x * y) ' * z ' * x * y * z = x * y @ z
    ]]></literal>
    <justification jstring="[para(7(a,1),10(a,1,2,2))].">
      <j1 rule="para" parents="7 10"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x * y * z * y * x * u = y * x * z * x * y * u
    ]]></literal>
    <justification jstring="[para(11(a,1),7(a,1,1)),demod(7(5),7(4),7(3),7(2),7(9),7(8),7(7))].">
      <j1 rule="para" parents="11 7"/>
      <j2 rule="demod" parents="7 7 7 7 7 7 7"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      x * y * y * x = y * x * x * y
    ]]></literal>
    <justification jstring="[para(8(a,1),11(a,1,2,2)),demod(8(6))].">
      <j1 rule="para" parents="8 11"/>
      <j2 rule="demod" parents="8"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x * y * (x * y) ' * y * x = y * x * e
    ]]></literal>
    <justification jstring="[para(9(a,1),11(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="9 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x ' ' * e = x
    ]]></literal>
    <justification jstring="[para(9(a,1),13(a,1,2))].">
      <j1 rule="para" parents="9 13"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x ' ' * (x @ y) = y ' * x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1,2))].">
      <j1 rule="para" parents="10 13"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x ' ' * y = x * y
    ]]></literal>
    <justification jstring="[para(13(a,1),13(a,1,2))].">
      <j1 rule="para" parents="13 13"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x ' * y * x = y * (y @ x)
    ]]></literal>
    <justification jstring="[back_demod(32),demod(37(4)),flip(a)].">
      <j1 rule="back_demod" parents="32"/>
      <j2 rule="demod" parents="37"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x * e = x
    ]]></literal>
    <justification jstring="[back_demod(29),demod(37(4))].">
      <j1 rule="back_demod" parents="29"/>
      <j2 rule="demod" parents="37"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      x * y * (x * y) ' * y * x = y * x
    ]]></literal>
    <justification jstring="[back_demod(25),demod(40(8))].">
      <j1 rule="back_demod" parents="25"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      x ' * y * x * (y @ z) = (x @ y ') * z ' * y * z
    ]]></literal>
    <justification jstring="[para(10(a,1),14(a,1,2,2,2)),demod(37(6))].">
      <j1 rule="para" parents="10 14"/>
      <j2 rule="demod" parents="37"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      (x ' @ y) * x = x * (y @ x)
    ]]></literal>
    <justification jstring="[para(10(a,1),14(a,1,2)),demod(37(4)),flip(a)].">
      <j1 rule="para" parents="10 14"/>
      <j2 rule="demod" parents="37"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[para(11(a,1),14(a,1,2,2)),demod(13(7),13(5)),flip(a)].">
      <j1 rule="para" parents="11 14"/>
      <j2 rule="demod" parents="13 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      (x ' @ y) * x * z = x * (y @ x) * z
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,2)),demod(37(5)),flip(a)].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="demod" parents="37"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      e ' = e
    ]]></literal>
    <justification jstring="[para(40(a,1),9(a,1))].">
      <j1 rule="para" parents="40 9"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      (x * y ') ' * y ' * x = x * y ' @ y
    ]]></literal>
    <justification jstring="[para(9(a,1),15(a,1,2,2,2)),demod(40(6))].">
      <j1 rule="para" parents="9 15"/>
      <j2 rule="demod" parents="40"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      (x * y) ' * y * x = x * y @ x
    ]]></literal>
    <justification jstring="[para(13(a,1),15(a,1,2))].">
      <j1 rule="para" parents="13 15"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      x * y ' @ y = x * y ' @ x
    ]]></literal>
    <justification jstring="[back_demod(65),demod(70(6)),flip(a)].">
      <j1 rule="back_demod" parents="65"/>
      <j2 rule="demod" parents="70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_demod(42),demod(70(4))].">
      <j1 rule="back_demod" parents="42"/>
      <j2 rule="demod" parents="70"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x * x ' = e
    ]]></literal>
    <justification jstring="[para(37(a,1),9(a,1))].">
      <j1 rule="para" parents="37 9"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x ' * y * x * y ' = x @ y '
    ]]></literal>
    <justification jstring="[para(37(a,1),10(a,1,2))].">
      <j1 rule="para" parents="37 10"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      x * x ' * y = y
    ]]></literal>
    <justification jstring="[para(37(a,1),13(a,1))].">
      <j1 rule="para" parents="37 13"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(37(a,1),40(a,1)),demod(40(2)),flip(a)].">
      <j1 rule="para" parents="37 40"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      x * y * (x * y) ' = e
    ]]></literal>
    <justification jstring="[para(79(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="79 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      x * y @ y ' = x * y @ x
    ]]></literal>
    <justification jstring="[para(79(a,1),15(a,1,2,2,2)),demod(88(4),40(4),70(4)),flip(a)].">
      <j1 rule="para" parents="79 15"/>
      <j2 rule="demod" parents="88 40 70"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      x ' * y ' * z * (y @ x) = y ' * x ' * z
    ]]></literal>
    <justification jstring="[para(10(a,1),19(a,1,2,2,2)),demod(13(12),9(10),40(10))].">
      <j1 rule="para" parents="10 19"/>
      <j2 rule="demod" parents="13 9 40"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      (x @ y) * z * y * x * u = z * x * y * u
    ]]></literal>
    <justification jstring="[para(19(a,1),14(a,1,2,2)),demod(13(8),13(6)),flip(a)].">
      <j1 rule="para" parents="19 14"/>
      <j2 rule="demod" parents="13 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="123">
    <literal><![CDATA[
      x * (y * x) ' = y '
    ]]></literal>
    <justification jstring="[para(91(a,1),13(a,1,2)),demod(40(3)),flip(a)].">
      <j1 rule="para" parents="91 13"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      (x @ y) * z * (x ' * y ' * x * y * z) ' = e
    ]]></literal>
    <justification jstring="[para(14(a,2),91(a,1,2,2,1))].">
      <j1 rule="para" parents="14 91"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[para(123(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="123 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="137">
    <literal><![CDATA[
      (x @ y) ' = y @ x
    ]]></literal>
    <justification jstring="[para(14(a,2),123(a,1,2,1)),demod(136(7),136(5),136(3),136(2),7(5),88(7),7(6),7(5),88(8),7(7),7(6),7(5),10(6),84(4)),flip(a)].">
      <j1 rule="para" parents="14 123"/>
      <j2 rule="demod" parents="136 136 136 136 7 88 7 7 88 7 7 7 10 84"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      (x @ y) * (y @ x) = e
    ]]></literal>
    <justification jstring="[back_demod(125),demod(136(8),136(6),136(4),136(3),7(6),88(8),7(7),7(6),88(9),7(8),7(7),7(6),10(7),84(5))].">
      <j1 rule="back_demod" parents="125"/>
      <j2 rule="demod" parents="136 136 136 136 7 88 7 7 88 7 7 7 10 84"/>
    </justification>
  </clause>

  <clause id="152">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[back_demod(70),demod(136(2),7(5),10(5)),flip(a)].">
      <j1 rule="back_demod" parents="70"/>
      <j2 rule="demod" parents="136 7 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      x * y @ y ' = y @ x
    ]]></literal>
    <justification jstring="[back_demod(94),demod(152(5))].">
      <j1 rule="back_demod" parents="94"/>
      <j2 rule="demod" parents="152"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[back_demod(76),demod(152(2))].">
      <j1 rule="back_demod" parents="76"/>
      <j2 rule="demod" parents="152"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      x * y ' @ y = y ' @ x
    ]]></literal>
    <justification jstring="[back_demod(75),demod(152(6))].">
      <j1 rule="back_demod" parents="75"/>
      <j2 rule="demod" parents="152"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      (x @ y) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(137(a,1),13(a,1,1))].">
      <j1 rule="para" parents="137 13"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      x * y @ x ' = y @ x '
    ]]></literal>
    <justification jstring="[para(13(a,1),152(a,1,1)),flip(a)].">
      <j1 rule="para" parents="13 152"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x ' * y @ x = y @ x
    ]]></literal>
    <justification jstring="[para(84(a,1),152(a,1,1)),flip(a)].">
      <j1 rule="para" parents="84 152"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="193">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[para(152(a,1),137(a,1,1)),demod(137(2)),flip(a)].">
      <j1 rule="para" parents="152 137"/>
      <j2 rule="demod" parents="137"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      x ' @ y ' * x * y = x ' @ (x @ y)
    ]]></literal>
    <justification jstring="[para(10(a,1),193(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 193"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="226">
    <literal><![CDATA[
      x ' @ y * x = y @ x
    ]]></literal>
    <justification jstring="[para(165(a,1),137(a,1,1)),demod(137(2)),flip(a)].">
      <j1 rule="para" parents="165 137"/>
      <j2 rule="demod" parents="137"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="240">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[para(23(a,1),14(a,1,2,2)),demod(13(6),13(4)),flip(a)].">
      <j1 rule="para" parents="23 14"/>
      <j2 rule="demod" parents="13 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="260">
    <literal><![CDATA[
      x @ y * x ' = y @ x '
    ]]></literal>
    <justification jstring="[para(88(a,1),226(a,1,1))].">
      <j1 rule="para" parents="88 226"/>
    </justification>
  </clause>

  <clause id="364">
    <literal><![CDATA[
      (x @ y) @ y * x = e
    ]]></literal>
    <justification jstring="[para(240(a,1),10(a,1,2,2)),demod(137(2),136(3),7(6),10(6),146(3)),flip(a)].">
      <j1 rule="para" parents="240 10"/>
      <j2 rule="demod" parents="137 136 7 10 146"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="366">
    <literal><![CDATA[
      (x ' @ (x @ y)) * (y @ x ') = x @ y
    ]]></literal>
    <justification jstring="[para(10(a,1),240(a,2)),demod(210(5),7(8),7(7),81(8))].">
      <j1 rule="para" parents="10 240"/>
      <j2 rule="demod" parents="210 7 7 81"/>
    </justification>
  </clause>

  <clause id="404">
    <literal><![CDATA[
      (x @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(84(a,1),364(a,1,2)),demod(191(3))].">
      <j1 rule="para" parents="84 364"/>
      <j2 rule="demod" parents="191"/>
    </justification>
  </clause>

  <clause id="422">
    <literal><![CDATA[
      x @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(404(a,1),137(a,1,1)),demod(62(2)),flip(a)].">
      <j1 rule="para" parents="404 137"/>
      <j2 rule="demod" parents="62"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="425">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[para(404(a,1),166(a,1,2,2)),demod(40(3)),flip(a)].">
      <j1 rule="para" parents="404 166"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="462">
    <literal><![CDATA[
      (x @ y) @ x ' = e
    ]]></literal>
    <justification jstring="[para(422(a,1),165(a,2)),demod(425(2),187(4))].">
      <j1 rule="para" parents="422 165"/>
      <j2 rule="demod" parents="425 187"/>
    </justification>
  </clause>

  <clause id="465">
    <literal><![CDATA[
      (x ' @ y) @ x = e
    ]]></literal>
    <justification jstring="[para(422(a,1),167(a,2)),demod(425(4),191(5))].">
      <j1 rule="para" parents="422 167"/>
      <j2 rule="demod" parents="425 191"/>
    </justification>
  </clause>

  <clause id="478">
    <literal><![CDATA[
      x ' @ (x @ y) = e
    ]]></literal>
    <justification jstring="[para(462(a,1),137(a,1,1)),demod(62(2)),flip(a)].">
      <j1 rule="para" parents="462 137"/>
      <j2 rule="demod" parents="62"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="483">
    <literal><![CDATA[
      (x @ y) @ y = e
    ]]></literal>
    <justification jstring="[para(226(a,1),462(a,1,1)),demod(88(3))].">
      <j1 rule="para" parents="226 462"/>
      <j2 rule="demod" parents="88"/>
    </justification>
  </clause>

  <clause id="486">
    <literal><![CDATA[
      x @ y ' = y @ x
    ]]></literal>
    <justification jstring="[back_demod(366),demod(478(3),8(4))].">
      <j1 rule="back_demod" parents="366"/>
      <j2 rule="demod" parents="478 8"/>
    </justification>
  </clause>

  <clause id="493">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(483(a,1),166(a,1,2,2)),demod(40(3)),flip(a)].">
      <j1 rule="para" parents="483 166"/>
      <j2 rule="demod" parents="40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="575">
    <literal><![CDATA[
      x ' @ y = y @ x
    ]]></literal>
    <justification jstring="[para(465(a,1),38(a,2,2)),demod(53(4),13(4),40(5)),flip(a)].">
      <j1 rule="para" parents="465 38"/>
      <j2 rule="demod" parents="53 13 40"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="583">
    <literal><![CDATA[
      (C @ A ') * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(486(a,2),12(a,1,1))].">
      <j1 rule="para" parents="486 12"/>
    </justification>
  </clause>

  <clause id="593">
    <literal><![CDATA[
      x ' @ y ' = x @ y
    ]]></literal>
    <justification jstring="[para(486(a,2),152(a,1)),demod(136(2),260(4))].">
      <j1 rule="para" parents="486 152"/>
      <j2 rule="demod" parents="136 260"/>
    </justification>
  </clause>

  <clause id="598">
    <literal><![CDATA[
      x * y @ z = z @ y ' * x '
    ]]></literal>
    <justification jstring="[para(136(a,1),486(a,1,2)),flip(a)].">
      <j1 rule="para" parents="136 486"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="602">
    <literal><![CDATA[
      (x @ y) * (x ' @ y) * z = z
    ]]></literal>
    <justification jstring="[para(486(a,1),180(a,1,1))].">
      <j1 rule="para" parents="486 180"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      C ' * A * C * A ' * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(14(a,2),583(a,1)),demod(88(5))].">
      <j1 rule="para" parents="14 583"/>
      <j2 rule="demod" parents="88"/>
    </justification>
  </clause>

  <clause id="701">
    <literal><![CDATA[
      x ' * (y @ x ') * z ' * x * z = (x @ y) * (x @ z)
    ]]></literal>
    <justification jstring="[para(51(a,1),14(a,1,2))].">
      <j1 rule="para" parents="51 14"/>
    </justification>
  </clause>

  <clause id="724">
    <literal><![CDATA[
      x ' * y * x * (y @ z) = (y @ x) * z ' * y * z
    ]]></literal>
    <justification jstring="[para(486(a,1),51(a,2,1))].">
      <j1 rule="para" parents="486 51"/>
    </justification>
  </clause>

  <clause id="1367">
    <literal><![CDATA[
      x @ (y @ x) * z = x @ z
    ]]></literal>
    <justification jstring="[para(61(a,2),10(a,1,2,2)),demod(136(4),137(4),7(9),602(8),10(5)),flip(a)].">
      <j1 rule="para" parents="61 10"/>
      <j2 rule="demod" parents="136 137 7 602 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1432">
    <literal><![CDATA[
      (x @ y) * x * z = x * (x @ y) * z
    ]]></literal>
    <justification jstring="[para(575(a,1),61(a,2,2,1)),demod(593(3))].">
      <j1 rule="para" parents="575 61"/>
      <j2 rule="demod" parents="593"/>
    </justification>
  </clause>

  <clause id="1537">
    <literal><![CDATA[
      (C @ A ' * (B @ C)) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(54(a,2),657(a,1,2)),demod(7(19),493(18),84(20),1432(16),13(17))].">
      <j1 rule="para" parents="54 657"/>
      <j2 rule="demod" parents="7 493 84 1432 13"/>
    </justification>
  </clause>

  <clause id="1645">
    <literal><![CDATA[
      (C @ A ' * (C @ B ')) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(486(a,2),1537(a,1,1,2,2))].">
      <j1 rule="para" parents="486 1537"/>
    </justification>
  </clause>

  <clause id="1855">
    <literal><![CDATA[
      (C @ A ' * (C @ B ')) * (C @ B ') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(486(a,2),1645(a,1,2))].">
      <j1 rule="para" parents="486 1645"/>
    </justification>
  </clause>

  <clause id="2412">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[para(99(a,2),14(a,1,2,2)),demod(88(2),88(2),84(7),84(5),593(5)),flip(a)].">
      <j1 rule="para" parents="99 14"/>
      <j2 rule="demod" parents="88 88 84 84 593"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2534">
    <literal><![CDATA[
      (C @ B ') * (C @ A ' * (C @ B ')) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_demod(1855),demod(2412(14,R))].">
      <j1 rule="back_demod" parents="1855"/>
      <j2 rule="demod" parents="2412"/>
    </justification>
  </clause>

  <clause id="3123">
    <literal><![CDATA[
      (x @ y) * z ' * x * z = x * (x @ y * z)
    ]]></literal>
    <justification jstring="[para(105(a,2),38(a,1)),demod(136(3),7(7),13(6))].">
      <j1 rule="para" parents="105 38"/>
      <j2 rule="demod" parents="136 7 13"/>
    </justification>
  </clause>

  <clause id="3209">
    <literal><![CDATA[
      x ' * y * x * (y @ z) = y * (y @ x * z)
    ]]></literal>
    <justification jstring="[back_demod(724),demod(3123(10))].">
      <j1 rule="back_demod" parents="724"/>
      <j2 rule="demod" parents="3123"/>
    </justification>
  </clause>

  <clause id="3226">
    <literal><![CDATA[
      (x @ y ') * z ' * y * z = y * (y @ x * z)
    ]]></literal>
    <justification jstring="[back_demod(51),demod(3209(5)),flip(a)].">
      <j1 rule="back_demod" parents="51"/>
      <j2 rule="demod" parents="3209"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3243">
    <literal><![CDATA[
      (x @ y) * (x @ z) = x @ y * z
    ]]></literal>
    <justification jstring="[back_demod(701),demod(3226(7),13(5)),flip(a)].">
      <j1 rule="back_demod" parents="701"/>
      <j2 rule="demod" parents="3226 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3246">
    <literal><![CDATA[
      C @ B ' * A ' * (C @ B ') != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_demod(2534),demod(3243(14))].">
      <j1 rule="back_demod" parents="2534"/>
      <j2 rule="demod" parents="3243"/>
    </justification>
  </clause>

  <clause id="3259">
    <literal><![CDATA[
      x @ (x @ y) * z = x @ z
    ]]></literal>
    <justification jstring="[para(84(a,1),1367(a,1,2)),demod(137(3)),flip(a)].">
      <j1 rule="para" parents="84 1367"/>
      <j2 rule="demod" parents="137"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3443">
    <literal><![CDATA[
      (x @ y) * (z @ u) * y * x = x * y * (z @ u)
    ]]></literal>
    <justification jstring="[para(2412(a,1),54(a,2)),demod(7(8))].">
      <j1 rule="para" parents="2412 54"/>
      <j2 rule="demod" parents="7"/>
    </justification>
  </clause>

  <clause id="3579">
    <literal><![CDATA[
      x @ y * z * (x @ u) = x @ y * z
    ]]></literal>
    <justification jstring="[para(54(a,2),3259(a,1,2)),demod(3443(5))].">
      <j1 rule="para" parents="54 3259"/>
      <j2 rule="demod" parents="3443"/>
    </justification>
  </clause>

  <clause id="3599">
    <literal><![CDATA[
      C @ B ' * A ' != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_demod(3246),demod(3579(12))].">
      <j1 rule="back_demod" parents="3246"/>
      <j2 rule="demod" parents="3579"/>
    </justification>
  </clause>

  <clause id="3600">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[resolve(3599,a,598,a(flip))].">
      <j1 rule="resolve" parents="3599 598"/>
    </justification>
  </clause>

</proof>

</proofs>
