<?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>EA-cs.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13125 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:35 2006
The command was "/home/mccune/bin/prover9 -f cs.in EA.in".
]]></heading>

<proof number="1" length="103" max_count="23">

<comments><![CDATA[
% Proof 1 at 4.79 (+ 0.05) seconds: A.
% Length of proof is 103.
% Level of proof is 31.
% Maximum clause weight is 23.
% Given clauses 791.
]]></comments>

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

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

  <clause id="9">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </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 @ B) @ C != D @ (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1)),demod(9(4))].">
      <j1 rule="para" parents="9 7"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * z != u * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

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

  <clause id="16">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="10 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x * y * (y @ z) != y * z
    ]]></literal>
    <literal><![CDATA[
      z = x
    ]]></literal>
    <justification jstring="[para(10(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="10 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1)),demod(9(2),9(5)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="demod" parents="9 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * y * z * (z @ x * y) = z * x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      ((A @ B) @ C) * x != (D @ (F @ G)) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(8,b,12,a)].">
      <j1 rule="ur" parents="8 12"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(15(a,1),7(a,1)),flip(a)].">
      <j1 rule="para" parents="15 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(15(a,1),9(a,1,1)),flip(a)].">
      <j1 rule="para" parents="15 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x * y * z != x * u * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ y) = z
    ]]></literal>
    <justification jstring="[para(10(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      x * y * z * y * x != y * x * u
    ]]></literal>
    <literal><![CDATA[
      z * x * y = u
    ]]></literal>
    <justification jstring="[para(11(a,1),13(a,1))].">
      <j1 rule="para" parents="11 13"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      x * y * z * u != v * u
    ]]></literal>
    <literal><![CDATA[
      x * y * z = v
    ]]></literal>
    <justification jstring="[para(9(a,1),14(a,1,2))].">
      <j1 rule="para" parents="9 14"/>
    </justification>
  </clause>

  <clause id="59">
    <literal><![CDATA[
      x * y * z != u * v * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u * v
    ]]></literal>
    <justification jstring="[para(9(a,1),14(a,2))].">
      <j1 rule="para" parents="9 14"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(13,a,38,a)].">
      <j1 rule="hyper" parents="13 38"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      z @ z = x
    ]]></literal>
    <justification jstring="[para(71(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="71 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(71(a,1),10(a,1,2)),demod(71(5))].">
      <j1 rule="para" parents="71 10"/>
      <j2 rule="demod" parents="71"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[hyper(16,a,9,a(flip)),demod(9(4))].">
      <j1 rule="hyper" parents="16 9"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x * y * z != u * x * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ x * y) = z
    ]]></literal>
    <justification jstring="[para(9(a,1),16(a,1))].">
      <j1 rule="para" parents="9 16"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(35,a,80,a),flip(a)].">
      <j1 rule="hyper" parents="35 80"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      (u @ u) @ y = z
    ]]></literal>
    <justification jstring="[para(80(a,1),13(a,1,2)),flip(a)].">
      <j1 rule="para" parents="80 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="127">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(98(a,2),71(a,1,1))].">
      <j1 rule="para" parents="98 71"/>
    </justification>
  </clause>

  <clause id="128">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(98(a,2),80(a,1,2,1))].">
      <j1 rule="para" parents="98 80"/>
    </justification>
  </clause>

  <clause id="137">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(16,a,127,a)].">
      <j1 rule="hyper" parents="16 127"/>
    </justification>
  </clause>

  <clause id="143">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(127(a,1),16(a,2)),demod(127(8))].">
      <j1 rule="para" parents="127 16"/>
      <j2 rule="demod" parents="127"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      x * (x @ y) != x * y
    ]]></literal>
    <literal><![CDATA[
      (z @ z) @ u = y
    ]]></literal>
    <justification jstring="[para(127(a,1),17(a,1)),flip(b)].">
      <j1 rule="para" parents="127 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(98(a,2),127(a,1,1,1))].">
      <j1 rule="para" parents="98 127"/>
    </justification>
  </clause>

  <clause id="149">
    <literal><![CDATA[
      x * y * z != y * u
    ]]></literal>
    <literal><![CDATA[
      x * (x @ y) * z = u
    ]]></literal>
    <justification jstring="[para(19(a,1),7(a,1))].">
      <j1 rule="para" parents="19 7"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      x * y * z * (z @ (x @ y)) = y * x * z * (x @ y)
    ]]></literal>
    <justification jstring="[para(10(a,1),19(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="10 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="172">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(79,a,127,a),flip(a)].">
      <j1 rule="hyper" parents="79 127"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="173">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(79,a,71,a)].">
      <j1 rule="hyper" parents="79 71"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(173)].">
      <j1 rule="new_symbol" parents="173"/>
    </justification>
  </clause>

  <clause id="181">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_demod(172),demod(177(1),177(3))].">
      <j1 rule="back_demod" parents="172"/>
      <j2 rule="demod" parents="177 177"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_demod(147),demod(177(1),181(2),181(2))].">
      <j1 rule="back_demod" parents="147"/>
      <j2 rule="demod" parents="177 181 181"/>
    </justification>
  </clause>

  <clause id="188">
    <literal><![CDATA[
      x * (x @ y) != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_demod(146),demod(177(5),181(6))].">
      <j1 rule="back_demod" parents="146"/>
      <j2 rule="demod" parents="177 181"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_demod(143),demod(177(3),181(4),181(4))].">
      <j1 rule="back_demod" parents="143"/>
      <j2 rule="demod" parents="177 181 181"/>
    </justification>
  </clause>

  <clause id="194">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_demod(137),demod(177(1),181(2),177(4),181(5))].">
      <j1 rule="back_demod" parents="137"/>
      <j2 rule="demod" parents="177 181 177 181"/>
    </justification>
  </clause>

  <clause id="197">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_demod(128),demod(177(1),181(2),181(2))].">
      <j1 rule="back_demod" parents="128"/>
      <j2 rule="demod" parents="177 181 181"/>
    </justification>
  </clause>

  <clause id="200">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = z
    ]]></literal>
    <justification jstring="[back_demod(106),demod(177(5),181(6))].">
      <j1 rule="back_demod" parents="106"/>
      <j2 rule="demod" parents="177 181"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_demod(194),demod(197(5))].">
      <j1 rule="back_demod" parents="194"/>
      <j2 rule="demod" parents="197"/>
    </justification>
  </clause>

  <clause id="203">
    <literal><![CDATA[
      x * y * (y @ y * x) = y * x
    ]]></literal>
    <justification jstring="[hyper(7,a,20,a)].">
      <j1 rule="hyper" parents="7 20"/>
    </justification>
  </clause>

  <clause id="252">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(190,a,202,a),flip(a)].">
      <j1 rule="hyper" parents="190 202"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      x @ y = c_0
    ]]></literal>
    <justification jstring="[para(10(a,1),200(a,1)),flip(b)].">
      <j1 rule="para" parents="10 200"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="328">
    <literal><![CDATA[
      x * (y * x @ y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(16,a,87,a),flip(a)].">
      <j1 rule="hyper" parents="16 87"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="358">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(16,a,203,a),flip(a)].">
      <j1 rule="hyper" parents="16 203"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="378">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      z = y
    ]]></literal>
    <justification jstring="[para(187(a,1),49(a,1,2)),demod(197(3),252(5),197(5))].">
      <j1 rule="para" parents="187 49"/>
      <j2 rule="demod" parents="197 252 197"/>
    </justification>
  </clause>

  <clause id="391">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[hyper(378,a,328,a),flip(a)].">
      <j1 rule="hyper" parents="378 328"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="402">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[hyper(51,a,19,a)].">
      <j1 rule="hyper" parents="51 19"/>
    </justification>
  </clause>

  <clause id="403">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[hyper(51,a,19,a(flip)),flip(a)].">
      <j1 rule="hyper" parents="51 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="407">
    <literal><![CDATA[
      x * (x @ y) @ y = x * y @ y
    ]]></literal>
    <justification jstring="[para(10(a,1),391(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 391"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="426">
    <literal><![CDATA[
      ((A @ B) @ C) * (F @ G) * D != D * (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(402(a,1),29(a,2))].">
      <j1 rule="para" parents="402 29"/>
    </justification>
  </clause>

  <clause id="465">
    <literal><![CDATA[
      x * y @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(402(a,1),305(a,2)),demod(9(3),10(3)),xx(a)].">
      <j1 rule="para" parents="402 305"/>
      <j2 rule="demod" parents="9 10"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="477">
    <literal><![CDATA[
      x * y @ (x @ y) = c_0
    ]]></literal>
    <justification jstring="[para(402(a,1),391(a,1,1)),demod(465(6))].">
      <j1 rule="para" parents="402 391"/>
      <j2 rule="demod" parents="465"/>
    </justification>
  </clause>

  <clause id="499">
    <literal><![CDATA[
      x * y * z * u != v * u * z
    ]]></literal>
    <literal><![CDATA[
      x * y * (z @ u) = v
    ]]></literal>
    <justification jstring="[para(402(a,1),58(a,1,2,2))].">
      <j1 rule="para" parents="402 58"/>
    </justification>
  </clause>

  <clause id="504">
    <literal><![CDATA[
      ((A @ B) @ C) * (F @ G) * D * x != D * (F @ G) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(58,b,426,a),demod(9(18))].">
      <j1 rule="ur" parents="58 426"/>
      <j2 rule="demod" parents="9"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      x * y @ (y @ x * (x @ y)) = c_0
    ]]></literal>
    <justification jstring="[para(10(a,1),477(a,1,1))].">
      <j1 rule="para" parents="10 477"/>
    </justification>
  </clause>

  <clause id="534">
    <literal><![CDATA[
      (x @ y * x) * y * x = x * y
    ]]></literal>
    <justification jstring="[hyper(59,a,402,a)].">
      <j1 rule="hyper" parents="59 402"/>
    </justification>
  </clause>

  <clause id="722">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(378,a,358,a),flip(a)].">
      <j1 rule="hyper" parents="378 358"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="740">
    <literal><![CDATA[
      x * (x @ y * x) = (x @ y) * x
    ]]></literal>
    <justification jstring="[hyper(88,a,19,a)].">
      <j1 rule="hyper" parents="88 19"/>
    </justification>
  </clause>

  <clause id="751">
    <literal><![CDATA[
      x @ y * (y @ x) = x @ y * x
    ]]></literal>
    <justification jstring="[para(10(a,1),722(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 722"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="778">
    <literal><![CDATA[
      x * y @ (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_demod(516),demod(751(4))].">
      <j1 rule="back_demod" parents="516"/>
      <j2 rule="demod" parents="751"/>
    </justification>
  </clause>

  <clause id="855">
    <literal><![CDATA[
      x * (y @ x) * y = y * x
    ]]></literal>
    <justification jstring="[para(778(a,1),10(a,1,2,2)),demod(197(5),534(4),9(5),740(4)),flip(a)].">
      <j1 rule="para" parents="778 10"/>
      <j2 rule="demod" parents="197 534 9 740"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="914">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(16,a,855,a),flip(a)].">
      <j1 rule="hyper" parents="16 855"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="959">
    <literal><![CDATA[
      x * (x @ y * x) = x * (x @ y)
    ]]></literal>
    <justification jstring="[back_demod(740),demod(914(5))].">
      <j1 rule="back_demod" parents="740"/>
      <j2 rule="demod" parents="914"/>
    </justification>
  </clause>

  <clause id="967">
    <literal><![CDATA[
      x * (x @ y) * (y @ x) = x
    ]]></literal>
    <justification jstring="[hyper(149,a,10,a)].">
      <j1 rule="hyper" parents="149 10"/>
    </justification>
  </clause>

  <clause id="1154">
    <literal><![CDATA[
      (x @ y) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(190,a,967,a),flip(a)].">
      <j1 rule="hyper" parents="190 967"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1155">
    <literal><![CDATA[
      (x @ y) * (y @ x) * x = (y @ x) * x * (x @ y)
    ]]></literal>
    <justification jstring="[para(967(a,1),11(a,1,2,2)),demod(914(3),1154(9),197(8)),flip(a)].">
      <j1 rule="para" parents="967 11"/>
      <j2 rule="demod" parents="914 1154 197"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1191">
    <literal><![CDATA[
      (x @ y) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(1154(a,1),9(a,1,1)),demod(187(2)),flip(a)].">
      <j1 rule="para" parents="1154 9"/>
      <j2 rule="demod" parents="187"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1239">
    <literal><![CDATA[
      ((A @ B) @ C) * (F @ G) * D * (G @ F) != D
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(1154(a,1),504(a,2,2)),demod(197(18))].">
      <j1 rule="para" parents="1154 504"/>
      <j2 rule="demod" parents="197"/>
    </justification>
  </clause>

  <clause id="1254">
    <literal><![CDATA[
      (x @ y) * y * (y @ x) = y
    ]]></literal>
    <justification jstring="[back_demod(1155),demod(1191(4)),flip(a)].">
      <j1 rule="back_demod" parents="1155"/>
      <j2 rule="demod" parents="1191"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1388">
    <literal><![CDATA[
      (x @ y) * z @ (y @ x) = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(1191(a,1),391(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1191 391"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1409">
    <literal><![CDATA[
      (x @ y) @ (y @ x) * z = (x @ y) @ z
    ]]></literal>
    <justification jstring="[para(1191(a,1),722(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1191 722"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1523">
    <literal><![CDATA[
      x * (x @ y) @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(1254(a,1),305(a,2)),demod(9(4),1154(3),197(2)),xx(a)].">
      <j1 rule="para" parents="1254 305"/>
      <j2 rule="demod" parents="9 1154 197"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1532">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(1254(a,1),391(a,1,1)),demod(1523(6))].">
      <j1 rule="para" parents="1254 391"/>
      <j2 rule="demod" parents="1523"/>
    </justification>
  </clause>

  <clause id="1841">
    <literal><![CDATA[
      x * (x @ y) @ y * (y @ x) = y * x @ y * (y @ x)
    ]]></literal>
    <justification jstring="[para(10(a,1),407(a,2,1)),demod(751(3),959(3))].">
      <j1 rule="para" parents="10 407"/>
      <j2 rule="demod" parents="751 959"/>
    </justification>
  </clause>

  <clause id="1849">
    <literal><![CDATA[
      x * (x @ y) @ x * y = x * x * y @ x * y
    ]]></literal>
    <justification jstring="[para(722(a,1),407(a,1,1,2))].">
      <j1 rule="para" parents="722 407"/>
    </justification>
  </clause>

  <clause id="2155">
    <literal><![CDATA[
      x * y @ x * (x @ y) = x * y @ x * x * y
    ]]></literal>
    <justification jstring="[para(722(a,1),751(a,1,2,2))].">
      <j1 rule="para" parents="722 751"/>
    </justification>
  </clause>

  <clause id="2175">
    <literal><![CDATA[
      x * x * y @ x * y = y * x @ y * y * x
    ]]></literal>
    <justification jstring="[para(751(a,1),751(a,1,2,2)),demod(959(5),1841(5),2155(4),10(9),1849(8)),flip(a)].">
      <j1 rule="para" parents="751 751"/>
      <j2 rule="demod" parents="959 1841 2155 10 1849"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2304">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[hyper(378,a,959,a),flip(a)].">
      <j1 rule="hyper" parents="378 959"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2323">
    <literal><![CDATA[
      x * x * y @ x * y = x @ y
    ]]></literal>
    <justification jstring="[back_demod(2175),demod(2304(8),391(6))].">
      <j1 rule="back_demod" parents="2175"/>
      <j2 rule="demod" parents="2304 391"/>
    </justification>
  </clause>

  <clause id="2326">
    <literal><![CDATA[
      x * y @ x * (x @ y) = y @ x
    ]]></literal>
    <justification jstring="[back_demod(2155),demod(2304(8),391(6))].">
      <j1 rule="back_demod" parents="2155"/>
      <j2 rule="demod" parents="2304 391"/>
    </justification>
  </clause>

  <clause id="2346">
    <literal><![CDATA[
      x * (x @ y) @ x * y = x @ y
    ]]></literal>
    <justification jstring="[back_demod(1849),demod(2323(8))].">
      <j1 rule="back_demod" parents="1849"/>
      <j2 rule="demod" parents="2323"/>
    </justification>
  </clause>

  <clause id="2349">
    <literal><![CDATA[
      x @ y * z * x = x @ y * z
    ]]></literal>
    <justification jstring="[para(9(a,1),2304(a,1,2))].">
      <j1 rule="para" parents="9 2304"/>
    </justification>
  </clause>

  <clause id="2350">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(10(a,1),2304(a,1,2)),demod(2346(4),407(4)),flip(a)].">
      <j1 rule="para" parents="10 2304"/>
      <j2 rule="demod" parents="2346 407"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2366">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(1191(a,1),2304(a,1,2)),demod(2350(3),1388(6))].">
      <j1 rule="para" parents="1191 2304"/>
      <j2 rule="demod" parents="2350 1388"/>
    </justification>
  </clause>

  <clause id="2557">
    <literal><![CDATA[
      (C @ (B @ A)) * (F @ G) * D * (G @ F) != D
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_demod(1239),demod(2366(5))].">
      <j1 rule="back_demod" parents="1239"/>
      <j2 rule="demod" parents="2366"/>
    </justification>
  </clause>

  <clause id="2850">
    <literal><![CDATA[
      (x @ y) @ ((y @ x) @ z) = c_0
    ]]></literal>
    <justification jstring="[para(2366(a,2),1532(a,1,2))].">
      <j1 rule="para" parents="2366 1532"/>
    </justification>
  </clause>

  <clause id="2925">
    <literal><![CDATA[
      x * ((y @ z) @ x) = (y @ z) * x * (z @ y)
    ]]></literal>
    <justification jstring="[para(2850(a,1),154(a,1,2,2,2)),demod(197(4),1191(10)),flip(a)].">
      <j1 rule="para" parents="2850 154"/>
      <j2 rule="demod" parents="197 1191"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3015">
    <literal><![CDATA[
      x * y @ y * y * x = x @ y
    ]]></literal>
    <justification jstring="[para(2326(a,1),2346(a,1,1,2)),demod(9(3),10(3),9(5),10(4),2326(8))].">
      <j1 rule="para" parents="2326 2346"/>
      <j2 rule="demod" parents="9 10 9 10 2326"/>
    </justification>
  </clause>

  <clause id="3537">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x) * z * (y @ x) * z * (x @ y)
    ]]></literal>
    <justification jstring="[para(1191(a,1),3015(a,1,1)),demod(9(6),9(7),1409(12)),flip(a)].">
      <j1 rule="para" parents="1191 3015"/>
      <j2 rule="demod" parents="9 9 1409"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4676">
    <literal><![CDATA[
      x * y * z @ (z @ y) = c_0
    ]]></literal>
    <justification jstring="[para(403(a,1),305(a,2)),demod(9(4),9(3),10(3)),xx(a)].">
      <j1 rule="para" parents="403 305"/>
      <j2 rule="demod" parents="9 9 10"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="5124">
    <literal><![CDATA[
      x * y @ ((z @ u) @ y) = c_0
    ]]></literal>
    <justification jstring="[para(1191(a,1),4676(a,1,1,2)),demod(2366(5,R),722(5))].">
      <j1 rule="para" parents="1191 4676"/>
      <j2 rule="demod" parents="2366 722"/>
    </justification>
  </clause>

  <clause id="5343">
    <literal><![CDATA[
      x * (y @ z) * u * (z @ y) != x * u
    ]]></literal>
    <literal><![CDATA[
      (y @ z) @ u = c_0
    ]]></literal>
    <justification jstring="[para(5124(a,1),188(a,1,2)),demod(197(3),9(5),2925(4)),flip(a),flip(b)].">
      <j1 rule="para" parents="5124 188"/>
      <j2 rule="demod" parents="197 9 2925"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9571">
    <literal><![CDATA[
      (x @ y) * z * (y @ x) = z
    ]]></literal>
    <justification jstring="[hyper(499,a,403,a)].">
      <j1 rule="hyper" parents="499 403"/>
    </justification>
  </clause>

  <clause id="9617">
    <literal><![CDATA[
      (x @ y) @ z = c_0
    ]]></literal>
    <justification jstring="[back_demod(5343),demod(9571(4)),xx(a)].">
      <j1 rule="back_demod" parents="5343"/>
      <j2 rule="demod" parents="9571"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="9657">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[back_demod(3537),demod(9617(2),9571(6),2349(5),2304(4)),flip(a)].">
      <j1 rule="back_demod" parents="3537"/>
      <j2 rule="demod" parents="9617 9571 2349 2304"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9660">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_demod(2557),demod(9657(5),9571(10),187(3)),xx(a)].">
      <j1 rule="back_demod" parents="2557"/>
      <j2 rule="demod" parents="9657 9571 187"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
