<?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="3">

<source>gtsax.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 12993 was started by mccune on cleo.thornwood,
Mon Jun 19 16:39:58 2006
The command was "/home/mccune/bin/prover9 -f gtsax.in".
]]></heading>

<proof number="1" length="11" max_count="29">

<comments><![CDATA[
% Proof 1 at 0.01 (+ 0.00) seconds: inverse.
% Length of proof is 11.
% Level of proof is 7.
% Maximum clause weight is 29.
% Given clauses 7.
]]></comments>

  <clause id="3">
    <literal><![CDATA[
      A * A ' != B * B '
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

  <clause id="7">
    <literal><![CDATA[
      B * B ' != A * A '
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[copy(3),flip(a)].">
      <j1 rule="copy" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x * ((y * (((z * z ') * (u * y) ') * v)) ' * (((w * w ') * u ') * x)) ' = v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,1,2,1,2,1))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

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

  <clause id="13">
    <literal><![CDATA[
      ((x * x ') * (y * z) ') * (u * u ') = v * ((y * (w * w ')) * (z * v)) '
    ]]></literal>
    <justification jstring="[para(10(a,1),5(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="10 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x * x ') * (y * (z * ((u * (v * v ')) * (y * z)) ')) ' = u
    ]]></literal>
    <justification jstring="[para(13(a,1),5(a,1,2,1,2))].">
      <j1 rule="para" parents="13 5"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x * (y * ((z * (u * u ')) * (x * y)) ')) ' * (v ' * z) ' = v
    ]]></literal>
    <justification jstring="[para(21(a,1),5(a,1,2,1,2))].">
      <j1 rule="para" parents="21 5"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      x * (((y * y ') * ((z * z ') * x) ') * u) = u
    ]]></literal>
    <justification jstring="[para(21(a,1),9(a,1,2,1,2)),demod(67(22))].">
      <j1 rule="para" parents="21 9"/>
      <j2 rule="demod" parents="67"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      x * x ' = y * y '
    ]]></literal>
    <justification jstring="[para(117(a,1),5(a,1,2,1))].">
      <j1 rule="para" parents="117 5"/>
    </justification>
  </clause>

  <clause id="126">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(inverse)
    ]]></attribute>
    <justification jstring="[resolve(125,a,7,a)].">
      <j1 rule="resolve" parents="125 7"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="55" max_count="35">

<comments><![CDATA[
% Proof 2 at 0.14 (+ 0.00) seconds: identity.
% Length of proof is 55.
% Level of proof is 24.
% Maximum clause weight is 35.
% Given clauses 25.
]]></comments>

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

  <clause id="8">
    <literal><![CDATA[
      A * (B * B ') != A
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      x * ((y * (((z * z ') * (u * y) ') * v)) ' * (((w * w ') * u ') * x)) ' = v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,1,2,1,2,1))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      (x * (((y * y ') * (z * x) ') * ((u * u ') * (v * w) '))) ' * (w * z) ' = v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,1,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x * x ') * (y * z) ') * (u * u ') = v * ((y * (w * w ')) * (z * v)) '
    ]]></literal>
    <justification jstring="[para(10(a,1),5(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="10 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x * x ') * (y * (z * ((u * (v * v ')) * (y * z)) ')) ' = u
    ]]></literal>
    <justification jstring="[para(13(a,1),5(a,1,2,1,2))].">
      <j1 rule="para" parents="13 5"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * ((y * ((z * (u * u ')) * (v * y)) ') * (z * x)) ' = v
    ]]></literal>
    <justification jstring="[para(13(a,1),10(a,1,2,1,1))].">
      <j1 rule="para" parents="13 10"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x * (y * ((z * (u * u ')) * (x * y)) ')) ' * (v ' * z) ' = v
    ]]></literal>
    <justification jstring="[para(21(a,1),5(a,1,2,1,2))].">
      <j1 rule="para" parents="21 5"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      x * (((y * y ') * ((z * z ') * x) ') * u) = u
    ]]></literal>
    <justification jstring="[para(21(a,1),9(a,1,2,1,2)),demod(67(22))].">
      <j1 rule="para" parents="21 9"/>
      <j2 rule="demod" parents="67"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      x * x ' = y * y '
    ]]></literal>
    <justification jstring="[para(117(a,1),5(a,1,2,1))].">
      <j1 rule="para" parents="117 5"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      (x * (((y * y ') * (z * x) ') * ((u * u ') * ((v * v ') * w) '))) ' = w * z
    ]]></literal>
    <justification jstring="[para(5(a,1),117(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5 117"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      x * (y ' * ((z * z ') * x)) ' = y
    ]]></literal>
    <justification jstring="[para(125(a,1),5(a,1,2,1,2,1))].">
      <j1 rule="para" parents="125 5"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      x * (((y * y ') * (z * z ')) * (u * x)) ' = u '
    ]]></literal>
    <justification jstring="[para(125(a,1),10(a,1,2,1,1,1))].">
      <j1 rule="para" parents="125 10"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      x ' * ((y * (z * z ')) * (u * u ')) ' = ((v * v ') * (y * x) ') * (w * w ')
    ]]></literal>
    <justification jstring="[para(125(a,1),13(a,2,2,1,2)),flip(a)].">
      <j1 rule="para" parents="125 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      (x * x ') ' * ((y * y ') * z) = z
    ]]></literal>
    <justification jstring="[para(125(a,1),117(a,1,2,1))].">
      <j1 rule="para" parents="125 117"/>
    </justification>
  </clause>

  <clause id="206">
    <literal><![CDATA[
      (x * x ') ' * (y ' * (z * z ')) ' = y
    ]]></literal>
    <justification jstring="[para(125(a,1),164(a,1,2,1,2))].">
      <j1 rule="para" parents="125 164"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      (x * x ') * y = z * ((((u * u ') * y ') * (v * v ')) * ((w * w ') ' * z)) '
    ]]></literal>
    <justification jstring="[para(180(a,1),10(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="180 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      ((x * x ') * y ') * (z * z ') = u * (((v * v ') ' * (w * w ')) * (((v6 * v6 ') * y) * u)) '
    ]]></literal>
    <justification jstring="[para(180(a,1),13(a,1,1,2,1))].">
      <j1 rule="para" parents="180 13"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      ((x * x ') * y) * ((z * (((u * u ') ' * (v * v ')) * (w * z)) ') * y) ' = w
    ]]></literal>
    <justification jstring="[para(180(a,1),29(a,1,2,1,2))].">
      <j1 rule="para" parents="180 29"/>
    </justification>
  </clause>

  <clause id="227">
    <literal><![CDATA[
      (x * x ') ' * y = ((z * z ') * ((u * u ') * (v * v ')) ') * y
    ]]></literal>
    <justification jstring="[para(117(a,1),180(a,1,2))].">
      <j1 rule="para" parents="117 180"/>
    </justification>
  </clause>

  <clause id="228">
    <literal><![CDATA[
      (x * x ') ' = (y * y ') ' * (z * z ')
    ]]></literal>
    <justification jstring="[para(125(a,1),180(a,1,2)),flip(a)].">
      <j1 rule="para" parents="125 180"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="300">
    <literal><![CDATA[
      (x * x ') * ((((y * y ') * ((z * z ') ' * u) ') * (v * v ')) * (w * w ') ') ' = u
    ]]></literal>
    <justification jstring="[para(228(a,2),10(a,1,2,1,2))].">
      <j1 rule="para" parents="228 10"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      (x * x ') ' = (y * y ') '
    ]]></literal>
    <justification jstring="[para(228(a,2),180(a,1))].">
      <j1 rule="para" parents="228 180"/>
    </justification>
  </clause>

  <clause id="356">
    <literal><![CDATA[
      (x * x ') * (y * y ') ' = z * z '
    ]]></literal>
    <justification jstring="[para(326(a,1),125(a,1,2))].">
      <j1 rule="para" parents="326 125"/>
    </justification>
  </clause>

  <clause id="386">
    <literal><![CDATA[
      ((x * x ') ' * y) * (z ' * y) ' = z
    ]]></literal>
    <justification jstring="[para(356(a,2),11(a,1,1,1,2,2,2,1)),demod(129(18))].">
      <j1 rule="para" parents="356 11"/>
      <j2 rule="demod" parents="129"/>
    </justification>
  </clause>

  <clause id="479">
    <literal><![CDATA[
      x * ((y * y ') ' ' * (z * x)) ' = z '
    ]]></literal>
    <justification jstring="[para(386(a,1),5(a,1,2,1,2,1))].">
      <j1 rule="para" parents="386 5"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      x * (y ' * (x ' * (z * z ')) ') ' = y
    ]]></literal>
    <justification jstring="[para(206(a,1),386(a,1,1))].">
      <j1 rule="para" parents="206 386"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      ((x * x ') ' * y ' ') * ((z * z ') * (u * u ') ') ' = y
    ]]></literal>
    <justification jstring="[para(356(a,2),386(a,1,2,1))].">
      <j1 rule="para" parents="356 386"/>
    </justification>
  </clause>

  <clause id="532">
    <literal><![CDATA[
      ((x * x ') * (y * (z * z ') ' ') ') ' = y
    ]]></literal>
    <justification jstring="[para(479(a,1),5(a,1))].">
      <j1 rule="para" parents="479 5"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      x ' * (y * y ') = z * (((u * u ') ' ' * (v * v ')) * ((x * (w * w ')) * z)) '
    ]]></literal>
    <justification jstring="[para(479(a,1),13(a,1,1))].">
      <j1 rule="para" parents="479 13"/>
    </justification>
  </clause>

  <clause id="582">
    <literal><![CDATA[
      (x * x ') * (y * z ') ' = z * y '
    ]]></literal>
    <justification jstring="[para(5(a,1),516(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="5 516"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="603">
    <literal><![CDATA[
      (x * x ') ' * y ' ' = y
    ]]></literal>
    <justification jstring="[para(180(a,1),516(a,1,2,1,2,1)),demod(582(10))].">
      <j1 rule="para" parents="180 516"/>
      <j2 rule="demod" parents="582"/>
    </justification>
  </clause>

  <clause id="622">
    <literal><![CDATA[
      ((x * x ') ' * y ') ' = y
    ]]></literal>
    <justification jstring="[back_demod(532),demod(582(9))].">
      <j1 rule="back_demod" parents="532"/>
      <j2 rule="demod" parents="582"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      x * (y * y ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(524),demod(603(6),582(6))].">
      <j1 rule="back_demod" parents="524"/>
      <j2 rule="demod" parents="603 582"/>
    </justification>
  </clause>

  <clause id="633">
    <literal><![CDATA[
      (x * x ') * (((y * y ') * ((z * z ') ' * u) ') * (v * v ')) ' = u
    ]]></literal>
    <justification jstring="[back_demod(300),demod(624(17))].">
      <j1 rule="back_demod" parents="300"/>
      <j2 rule="demod" parents="624"/>
    </justification>
  </clause>

  <clause id="769">
    <literal><![CDATA[
      x * ((((y * y ') * z ') * (u * u ')) * ((v * v ') ' * x)) ' = z ' '
    ]]></literal>
    <justification jstring="[para(603(a,1),10(a,1,2,1,1,1,2,1))].">
      <j1 rule="para" parents="603 10"/>
    </justification>
  </clause>

  <clause id="787">
    <literal><![CDATA[
      x ' ' = (y * y ') * x
    ]]></literal>
    <justification jstring="[back_demod(210),demod(769(17)),flip(a)].">
      <j1 rule="back_demod" parents="210"/>
      <j2 rule="demod" parents="769"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="847">
    <literal><![CDATA[
      ((x * x ') * y ' ' ') * (z * z ') = y '
    ]]></literal>
    <justification jstring="[para(787(a,2),13(a,1,1,2,1)),demod(167(18))].">
      <j1 rule="para" parents="787 13"/>
      <j2 rule="demod" parents="167"/>
    </justification>
  </clause>

  <clause id="882">
    <literal><![CDATA[
      x * (((y * y ') * x) ' ' ' * z) = z
    ]]></literal>
    <justification jstring="[para(787(a,2),117(a,1,2,1))].">
      <j1 rule="para" parents="787 117"/>
    </justification>
  </clause>

  <clause id="946">
    <literal><![CDATA[
      ((x * x ') * y) * (z * z ') = u * (((v * v ') ' * (w * w ')) * (y ' * u)) '
    ]]></literal>
    <justification jstring="[para(622(a,1),13(a,1,1,2))].">
      <j1 rule="para" parents="622 13"/>
    </justification>
  </clause>

  <clause id="964">
    <literal><![CDATA[
      x ' * (y * y ') = x '
    ]]></literal>
    <justification jstring="[para(164(a,1),622(a,1,1)),demod(624(8)),flip(a)].">
      <j1 rule="para" parents="164 622"/>
      <j2 rule="demod" parents="624"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="995">
    <literal><![CDATA[
      ((x * x ') * y) * (z * z ') = u * ((v * v ') ' * (y ' * u)) '
    ]]></literal>
    <justification jstring="[back_demod(946),demod(964(12))].">
      <j1 rule="back_demod" parents="946"/>
      <j2 rule="demod" parents="964"/>
    </justification>
  </clause>

  <clause id="1004">
    <literal><![CDATA[
      (x * (y * y ')) ' = x '
    ]]></literal>
    <justification jstring="[back_demod(537),demod(964(4),964(8),479(12)),flip(a)].">
      <j1 rule="back_demod" parents="537"/>
      <j2 rule="demod" parents="964 964 479"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1009">
    <literal><![CDATA[
      ((x * x ') * y) * ((z * ((u * u ') ' * (v * z)) ') * y) ' = v
    ]]></literal>
    <justification jstring="[back_demod(220),demod(964(9))].">
      <j1 rule="back_demod" parents="220"/>
      <j2 rule="demod" parents="964"/>
    </justification>
  </clause>

  <clause id="1010">
    <literal><![CDATA[
      ((x * x ') * y ') * (z * z ') = u * ((v * v ') ' * (((w * w ') * y) * u)) '
    ]]></literal>
    <justification jstring="[back_demod(213),demod(964(13))].">
      <j1 rule="back_demod" parents="213"/>
      <j2 rule="demod" parents="964"/>
    </justification>
  </clause>

  <clause id="1032">
    <literal><![CDATA[
      (x * x ') ' * y = y
    ]]></literal>
    <justification jstring="[back_demod(633),demod(1004(14),582(12),624(8))].">
      <j1 rule="back_demod" parents="633"/>
      <j2 rule="demod" parents="1004 582 624"/>
    </justification>
  </clause>

  <clause id="1047">
    <literal><![CDATA[
      (x * x ') * y = y
    ]]></literal>
    <justification jstring="[back_demod(227),demod(1032(4),1004(8),624(6)),flip(a)].">
      <j1 rule="back_demod" parents="227"/>
      <j2 rule="demod" parents="1032 1004 624"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1053">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[back_demod(171),demod(1004(8),1004(5),1047(8),964(8)),flip(a)].">
      <j1 rule="back_demod" parents="171"/>
      <j2 rule="demod" parents="1004 1004 1047 964"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1070">
    <literal><![CDATA[
      x ' ' ' = x '
    ]]></literal>
    <justification jstring="[back_demod(847),demod(1047(6),964(6))].">
      <j1 rule="back_demod" parents="847"/>
      <j2 rule="demod" parents="1047 964"/>
    </justification>
  </clause>

  <clause id="1106">
    <literal><![CDATA[
      x * (x ' * y) = y
    ]]></literal>
    <justification jstring="[back_demod(882),demod(1047(3),1070(3))].">
      <j1 rule="back_demod" parents="882"/>
      <j2 rule="demod" parents="1047 1070"/>
    </justification>
  </clause>

  <clause id="1117">
    <literal><![CDATA[
      x * ((x ' * y ') * (z ' ' * z ')) = y '
    ]]></literal>
    <justification jstring="[back_demod(1010),demod(1047(4),964(4),1053(4),1047(8),1053(8),1053(3),1053(9),1070(9)),flip(a)].">
      <j1 rule="back_demod" parents="1010"/>
      <j2 rule="demod" parents="1047 964 1053 1047 1053 1053 1053 1070"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1118">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(1009),demod(1047(3),1053(3),1053(7),1053(2),1053(8),1070(8),1117(9),1053(3),1106(5))].">
      <j1 rule="back_demod" parents="1009"/>
      <j2 rule="demod" parents="1047 1053 1053 1053 1053 1070 1117 1053 1106"/>
    </justification>
  </clause>

  <clause id="1119">
    <literal><![CDATA[
      x * (y * y ') = x
    ]]></literal>
    <justification jstring="[back_demod(995),demod(1047(3),1053(6),1118(5),1047(8),1053(6),1118(6),1106(6))].">
      <j1 rule="back_demod" parents="995"/>
      <j2 rule="demod" parents="1047 1053 1118 1047 1053 1118 1106"/>
    </justification>
  </clause>

  <clause id="1120">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(identity)
    ]]></attribute>
    <justification jstring="[resolve(1119,a,8,a)].">
      <j1 rule="resolve" parents="1119 8"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="61" max_count="38">

<comments><![CDATA[
% Proof 3 at 0.19 (+ 0.00) seconds: associativity.
% Length of proof is 61.
% Level of proof is 25.
% Maximum clause weight is 38.
% Given clauses 25.
]]></comments>

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

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

  <clause id="9">
    <literal><![CDATA[
      x * ((y * (((z * z ') * (u * y) ') * v)) ' * (((w * w ') * u ') * x)) ' = v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,1,2,1,2,1))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      (x * (((y * y ') * (z * x) ') * ((u * u ') * (v * w) '))) ' * (w * z) ' = v
    ]]></literal>
    <justification jstring="[para(5(a,1),5(a,1,2,1,2))].">
      <j1 rule="para" parents="5 5"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x * x ') * (y * z) ') * (u * u ') = v * ((y * (w * w ')) * (z * v)) '
    ]]></literal>
    <justification jstring="[para(10(a,1),5(a,1,2,1,2,1)),flip(a)].">
      <j1 rule="para" parents="10 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (x * x ') * (y * (z * ((u * (v * v ')) * (y * z)) ')) ' = u
    ]]></literal>
    <justification jstring="[para(13(a,1),5(a,1,2,1,2))].">
      <j1 rule="para" parents="13 5"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * ((y * ((z * (u * u ')) * (v * y)) ') * (z * x)) ' = v
    ]]></literal>
    <justification jstring="[para(13(a,1),10(a,1,2,1,1))].">
      <j1 rule="para" parents="13 10"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (x * (y * ((z * (u * u ')) * (x * y)) ')) ' * (v ' * z) ' = v
    ]]></literal>
    <justification jstring="[para(21(a,1),5(a,1,2,1,2))].">
      <j1 rule="para" parents="21 5"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      x * (((y * y ') * ((z * z ') * x) ') * u) = u
    ]]></literal>
    <justification jstring="[para(21(a,1),9(a,1,2,1,2)),demod(67(22))].">
      <j1 rule="para" parents="21 9"/>
      <j2 rule="demod" parents="67"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      x * x ' = y * y '
    ]]></literal>
    <justification jstring="[para(117(a,1),5(a,1,2,1))].">
      <j1 rule="para" parents="117 5"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      (x * (((y * y ') * (z * x) ') * ((u * u ') * ((v * v ') * w) '))) ' = w * z
    ]]></literal>
    <justification jstring="[para(5(a,1),117(a,1,2)),flip(a)].">
      <j1 rule="para" parents="5 117"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      x * (y ' * ((z * z ') * x)) ' = y
    ]]></literal>
    <justification jstring="[para(125(a,1),5(a,1,2,1,2,1))].">
      <j1 rule="para" parents="125 5"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      x * (((y * y ') * (z * z ')) * (u * x)) ' = u '
    ]]></literal>
    <justification jstring="[para(125(a,1),10(a,1,2,1,1,1))].">
      <j1 rule="para" parents="125 10"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      x ' * ((y * (z * z ')) * (u * u ')) ' = ((v * v ') * (y * x) ') * (w * w ')
    ]]></literal>
    <justification jstring="[para(125(a,1),13(a,2,2,1,2)),flip(a)].">
      <j1 rule="para" parents="125 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      ((x * x ') * (y * z) ') ' = u * ((z * (v * v ')) ' * (((w * w ') * y ') * u)) '
    ]]></literal>
    <justification jstring="[para(125(a,1),9(a,1,2,1,1,1,2)),flip(a)].">
      <j1 rule="para" parents="125 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      (x * x ') ' * ((y * y ') * z) = z
    ]]></literal>
    <justification jstring="[para(125(a,1),117(a,1,2,1))].">
      <j1 rule="para" parents="125 117"/>
    </justification>
  </clause>

  <clause id="206">
    <literal><![CDATA[
      (x * x ') ' * (y ' * (z * z ')) ' = y
    ]]></literal>
    <justification jstring="[para(125(a,1),164(a,1,2,1,2))].">
      <j1 rule="para" parents="125 164"/>
    </justification>
  </clause>

  <clause id="210">
    <literal><![CDATA[
      (x * x ') * y = z * ((((u * u ') * y ') * (v * v ')) * ((w * w ') ' * z)) '
    ]]></literal>
    <justification jstring="[para(180(a,1),10(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="180 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      ((x * x ') * y ') * (z * z ') = u * (((v * v ') ' * (w * w ')) * (((v6 * v6 ') * y) * u)) '
    ]]></literal>
    <justification jstring="[para(180(a,1),13(a,1,1,2,1))].">
      <j1 rule="para" parents="180 13"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      ((x * x ') * y) * ((z * (((u * u ') ' * (v * v ')) * (w * z)) ') * y) ' = w
    ]]></literal>
    <justification jstring="[para(180(a,1),29(a,1,2,1,2))].">
      <j1 rule="para" parents="180 29"/>
    </justification>
  </clause>

  <clause id="227">
    <literal><![CDATA[
      (x * x ') ' * y = ((z * z ') * ((u * u ') * (v * v ')) ') * y
    ]]></literal>
    <justification jstring="[para(117(a,1),180(a,1,2))].">
      <j1 rule="para" parents="117 180"/>
    </justification>
  </clause>

  <clause id="228">
    <literal><![CDATA[
      (x * x ') ' = (y * y ') ' * (z * z ')
    ]]></literal>
    <justification jstring="[para(125(a,1),180(a,1,2)),flip(a)].">
      <j1 rule="para" parents="125 180"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="279">
    <literal><![CDATA[
      (x ' * (y * y ')) ' = z * ((((u * u ') * x ') * (v * v ')) * ((w * w ') ' * z)) '
    ]]></literal>
    <justification jstring="[para(206(a,1),10(a,1,2,1,1,1,2,1)),flip(a)].">
      <j1 rule="para" parents="206 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="300">
    <literal><![CDATA[
      (x * x ') * ((((y * y ') * ((z * z ') ' * u) ') * (v * v ')) * (w * w ') ') ' = u
    ]]></literal>
    <justification jstring="[para(228(a,2),10(a,1,2,1,2))].">
      <j1 rule="para" parents="228 10"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      (x * x ') ' = (y * y ') '
    ]]></literal>
    <justification jstring="[para(228(a,2),180(a,1))].">
      <j1 rule="para" parents="228 180"/>
    </justification>
  </clause>

  <clause id="356">
    <literal><![CDATA[
      (x * x ') * (y * y ') ' = z * z '
    ]]></literal>
    <justification jstring="[para(326(a,1),125(a,1,2))].">
      <j1 rule="para" parents="326 125"/>
    </justification>
  </clause>

  <clause id="386">
    <literal><![CDATA[
      ((x * x ') ' * y) * (z ' * y) ' = z
    ]]></literal>
    <justification jstring="[para(356(a,2),11(a,1,1,1,2,2,2,1)),demod(129(18))].">
      <j1 rule="para" parents="356 11"/>
      <j2 rule="demod" parents="129"/>
    </justification>
  </clause>

  <clause id="479">
    <literal><![CDATA[
      x * ((y * y ') ' ' * (z * x)) ' = z '
    ]]></literal>
    <justification jstring="[para(386(a,1),5(a,1,2,1,2,1))].">
      <j1 rule="para" parents="386 5"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      x * (y ' * (x ' * (z * z ')) ') ' = y
    ]]></literal>
    <justification jstring="[para(206(a,1),386(a,1,1))].">
      <j1 rule="para" parents="206 386"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      ((x * x ') ' * y ' ') * ((z * z ') * (u * u ') ') ' = y
    ]]></literal>
    <justification jstring="[para(356(a,2),386(a,1,2,1))].">
      <j1 rule="para" parents="356 386"/>
    </justification>
  </clause>

  <clause id="532">
    <literal><![CDATA[
      ((x * x ') * (y * (z * z ') ' ') ') ' = y
    ]]></literal>
    <justification jstring="[para(479(a,1),5(a,1))].">
      <j1 rule="para" parents="479 5"/>
    </justification>
  </clause>

  <clause id="537">
    <literal><![CDATA[
      x ' * (y * y ') = z * (((u * u ') ' ' * (v * v ')) * ((x * (w * w ')) * z)) '
    ]]></literal>
    <justification jstring="[para(479(a,1),13(a,1,1))].">
      <j1 rule="para" parents="479 13"/>
    </justification>
  </clause>

  <clause id="582">
    <literal><![CDATA[
      (x * x ') * (y * z ') ' = z * y '
    ]]></literal>
    <justification jstring="[para(5(a,1),516(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="5 516"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="594">
    <literal><![CDATA[
      (x ' * (((y * y ') * (z * u) ') ' * (v * v ')) ') ' = w * ((u * x) ' * (((v6 * v6 ') * z ') * w)) '
    ]]></literal>
    <justification jstring="[para(516(a,1),9(a,1,2,1,1,1,2)),flip(a)].">
      <j1 rule="para" parents="516 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="603">
    <literal><![CDATA[
      (x * x ') ' * y ' ' = y
    ]]></literal>
    <justification jstring="[para(180(a,1),516(a,1,2,1,2,1)),demod(582(10))].">
      <j1 rule="para" parents="180 516"/>
      <j2 rule="demod" parents="582"/>
    </justification>
  </clause>

  <clause id="622">
    <literal><![CDATA[
      ((x * x ') ' * y ') ' = y
    ]]></literal>
    <justification jstring="[back_demod(532),demod(582(9))].">
      <j1 rule="back_demod" parents="532"/>
      <j2 rule="demod" parents="582"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      x * (y * y ') ' = x
    ]]></literal>
    <justification jstring="[back_demod(524),demod(603(6),582(6))].">
      <j1 rule="back_demod" parents="524"/>
      <j2 rule="demod" parents="603 582"/>
    </justification>
  </clause>

  <clause id="633">
    <literal><![CDATA[
      (x * x ') * (((y * y ') * ((z * z ') ' * u) ') * (v * v ')) ' = u
    ]]></literal>
    <justification jstring="[back_demod(300),demod(624(17))].">
      <j1 rule="back_demod" parents="300"/>
      <j2 rule="demod" parents="624"/>
    </justification>
  </clause>

  <clause id="769">
    <literal><![CDATA[
      x * ((((y * y ') * z ') * (u * u ')) * ((v * v ') ' * x)) ' = z ' '
    ]]></literal>
    <justification jstring="[para(603(a,1),10(a,1,2,1,1,1,2,1))].">
      <j1 rule="para" parents="603 10"/>
    </justification>
  </clause>

  <clause id="786">
    <literal><![CDATA[
      (x ' * (y * y ')) ' = x ' '
    ]]></literal>
    <justification jstring="[back_demod(279),demod(769(19))].">
      <j1 rule="back_demod" parents="279"/>
      <j2 rule="demod" parents="769"/>
    </justification>
  </clause>

  <clause id="787">
    <literal><![CDATA[
      x ' ' = (y * y ') * x
    ]]></literal>
    <justification jstring="[back_demod(210),demod(769(17)),flip(a)].">
      <j1 rule="back_demod" parents="210"/>
      <j2 rule="demod" parents="769"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="801">
    <literal><![CDATA[
      (x ' * ((y * y ') * (z * u) ') ' ') ' = v * ((u * x) ' * (((w * w ') * z ') * v)) '
    ]]></literal>
    <justification jstring="[back_demod(594),demod(786(11))].">
      <j1 rule="back_demod" parents="594"/>
      <j2 rule="demod" parents="786"/>
    </justification>
  </clause>

  <clause id="847">
    <literal><![CDATA[
      ((x * x ') * y ' ' ') * (z * z ') = y '
    ]]></literal>
    <justification jstring="[para(787(a,2),13(a,1,1,2,1)),demod(167(18))].">
      <j1 rule="para" parents="787 13"/>
      <j2 rule="demod" parents="167"/>
    </justification>
  </clause>

  <clause id="882">
    <literal><![CDATA[
      x * (((y * y ') * x) ' ' ' * z) = z
    ]]></literal>
    <justification jstring="[para(787(a,2),117(a,1,2,1))].">
      <j1 rule="para" parents="787 117"/>
    </justification>
  </clause>

  <clause id="964">
    <literal><![CDATA[
      x ' * (y * y ') = x '
    ]]></literal>
    <justification jstring="[para(164(a,1),622(a,1,1)),demod(624(8)),flip(a)].">
      <j1 rule="para" parents="164 622"/>
      <j2 rule="demod" parents="624"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1004">
    <literal><![CDATA[
      (x * (y * y ')) ' = x '
    ]]></literal>
    <justification jstring="[back_demod(537),demod(964(4),964(8),479(12)),flip(a)].">
      <j1 rule="back_demod" parents="537"/>
      <j2 rule="demod" parents="964 964 479"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1009">
    <literal><![CDATA[
      ((x * x ') * y) * ((z * ((u * u ') ' * (v * z)) ') * y) ' = v
    ]]></literal>
    <justification jstring="[back_demod(220),demod(964(9))].">
      <j1 rule="back_demod" parents="220"/>
      <j2 rule="demod" parents="964"/>
    </justification>
  </clause>

  <clause id="1010">
    <literal><![CDATA[
      ((x * x ') * y ') * (z * z ') = u * ((v * v ') ' * (((w * w ') * y) * u)) '
    ]]></literal>
    <justification jstring="[back_demod(213),demod(964(13))].">
      <j1 rule="back_demod" parents="213"/>
      <j2 rule="demod" parents="964"/>
    </justification>
  </clause>

  <clause id="1032">
    <literal><![CDATA[
      (x * x ') ' * y = y
    ]]></literal>
    <justification jstring="[back_demod(633),demod(1004(14),582(12),624(8))].">
      <j1 rule="back_demod" parents="633"/>
      <j2 rule="demod" parents="1004 582 624"/>
    </justification>
  </clause>

  <clause id="1047">
    <literal><![CDATA[
      (x * x ') * y = y
    ]]></literal>
    <justification jstring="[back_demod(227),demod(1032(4),1004(8),624(6)),flip(a)].">
      <j1 rule="back_demod" parents="227"/>
      <j2 rule="demod" parents="1032 1004 624"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1049">
    <literal><![CDATA[
      (x * y) ' ' = z * (y ' * (x ' * z)) '
    ]]></literal>
    <justification jstring="[back_demod(177),demod(1047(5),1004(7),1047(8))].">
      <j1 rule="back_demod" parents="177"/>
      <j2 rule="demod" parents="1047 1004 1047"/>
    </justification>
  </clause>

  <clause id="1053">
    <literal><![CDATA[
      (x * y) ' = y ' * x '
    ]]></literal>
    <justification jstring="[back_demod(171),demod(1004(8),1004(5),1047(8),964(8)),flip(a)].">
      <j1 rule="back_demod" parents="171"/>
      <j2 rule="demod" parents="1004 1004 1047 964"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1070">
    <literal><![CDATA[
      x ' ' ' = x '
    ]]></literal>
    <justification jstring="[back_demod(847),demod(1047(6),964(6))].">
      <j1 rule="back_demod" parents="847"/>
      <j2 rule="demod" parents="1047 964"/>
    </justification>
  </clause>

  <clause id="1106">
    <literal><![CDATA[
      x * (x ' * y) = y
    ]]></literal>
    <justification jstring="[back_demod(882),demod(1047(3),1070(3))].">
      <j1 rule="back_demod" parents="882"/>
      <j2 rule="demod" parents="1047 1070"/>
    </justification>
  </clause>

  <clause id="1109">
    <literal><![CDATA[
      (x ' ' * y ' ') * z ' ' = u * ((u ' * x ' ') * (y ' ' * z ' '))
    ]]></literal>
    <justification jstring="[back_demod(801),demod(1053(5),1047(7),1053(5),1053(7),1070(4),1070(5),1053(6),1053(4),1053(10),1047(15),1053(15),1053(11),1053(16))].">
      <j1 rule="back_demod" parents="801"/>
      <j2 rule="demod" parents="1053 1047 1053 1053 1070 1070 1053 1053 1053 1047 1053 1053 1053"/>
    </justification>
  </clause>

  <clause id="1117">
    <literal><![CDATA[
      x * ((x ' * y ') * (z ' ' * z ')) = y '
    ]]></literal>
    <justification jstring="[back_demod(1010),demod(1047(4),964(4),1053(4),1047(8),1053(8),1053(3),1053(9),1070(9)),flip(a)].">
      <j1 rule="back_demod" parents="1010"/>
      <j2 rule="demod" parents="1047 964 1053 1047 1053 1053 1053 1070"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1118">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[back_demod(1009),demod(1047(3),1053(3),1053(7),1053(2),1053(8),1070(8),1117(9),1053(3),1106(5))].">
      <j1 rule="back_demod" parents="1009"/>
      <j2 rule="demod" parents="1047 1053 1053 1053 1053 1070 1117 1053 1106"/>
    </justification>
  </clause>

  <clause id="1211">
    <literal><![CDATA[
      x * ((x ' * y) * z) = y * z
    ]]></literal>
    <justification jstring="[back_demod(1049),demod(1053(2),1053(4),1118(2),1118(2),1053(6),1053(4),1118(4),1118(5)),flip(a)].">
      <j1 rule="back_demod" parents="1049"/>
      <j2 rule="demod" parents="1053 1053 1118 1118 1053 1053 1118 1118"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1213">
    <literal><![CDATA[
      (x * y) * z = x * (y * z)
    ]]></literal>
    <justification jstring="[back_demod(1109),demod(1118(2),1118(2),1118(3),1118(5),1118(6),1118(6),1211(7))].">
      <j1 rule="back_demod" parents="1109"/>
      <j2 rule="demod" parents="1118 1118 1118 1118 1118 1118 1211"/>
    </justification>
  </clause>

  <clause id="1214">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(associativity)
    ]]></attribute>
    <justification jstring="[resolve(1213,a,6,a)].">
      <j1 rule="resolve" parents="1213 6"/>
    </justification>
  </clause>

</proof>

</proofs>
