<?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 November-2006, November 2006.
Process 3865 was started by mccune on cleo.thornwood,
Wed Nov 22 11:24:17 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 4.78 (+ 0.06) seconds: A.
% Length of proof is 103.
% Level of proof is 31.
% Maximum clause weight is 23.
% Given clauses 791.
]]></comments>

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

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

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

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

  <clause id="5" type="assumption">
    <literal><![CDATA[
      x * y * z * y * x = y * x * z * x * y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      (A @ B) @ C != D @ (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

  <clause id="9">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

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

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

  <clause id="13">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,1)),rewrite(3(2),3(5)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * z * (z @ x * y) = z * x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      ((A @ B) @ C) * x != (D @ (F @ G)) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(2,b,6,a)].">
      <j1 rule="ur" parents="2 6"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(9(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="9 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(9(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="9 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x * y * z != x * u * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ y) = z
    ]]></literal>
    <justification jstring="[para(4(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x * y * z * y * x != y * x * u
    ]]></literal>
    <literal><![CDATA[
      z * x * y = u
    ]]></literal>
    <justification jstring="[para(5(a,1),7(a,1))].">
      <j1 rule="para" parents="5 7"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      x * y * z * u != v * u
    ]]></literal>
    <literal><![CDATA[
      x * y * z = v
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,2))].">
      <j1 rule="para" parents="3 8"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      x * y * z != u * v * z
    ]]></literal>
    <literal><![CDATA[
      x * y = u * v
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,2))].">
      <j1 rule="para" parents="3 8"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(7,a,32,a)].">
      <j1 rule="hyper" parents="7 32"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      z @ z = x
    ]]></literal>
    <justification jstring="[para(65(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="65 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(65(a,1),4(a,1,2)),rewrite(65(5))].">
      <j1 rule="para" parents="65 4"/>
      <j2 rule="rewrite" parents="65"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x * y * (x * y @ x) = y * x
    ]]></literal>
    <justification jstring="[hyper(10,a,3,a(flip)),rewrite(3(4))].">
      <j1 rule="hyper" parents="10 3"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x * y * z != u * x * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ x * y) = z
    ]]></literal>
    <justification jstring="[para(3(a,1),10(a,1))].">
      <j1 rule="para" parents="3 10"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(29,a,74,a),flip(a)].">
      <j1 rule="hyper" parents="29 74"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      (u @ u) @ y = z
    ]]></literal>
    <justification jstring="[para(74(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="74 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="121">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(92(a,2),65(a,1,1))].">
      <j1 rule="para" parents="92 65"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(92(a,2),74(a,1,2,1))].">
      <j1 rule="para" parents="92 74"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(10,a,121,a)].">
      <j1 rule="hyper" parents="10 121"/>
    </justification>
  </clause>

  <clause id="137">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(121(a,1),10(a,2)),rewrite(121(8))].">
      <j1 rule="para" parents="121 10"/>
      <j2 rule="rewrite" parents="121"/>
    </justification>
  </clause>

  <clause id="140">
    <literal><![CDATA[
      x * (x @ y) != x * y
    ]]></literal>
    <literal><![CDATA[
      (z @ z) @ u = y
    ]]></literal>
    <justification jstring="[para(121(a,1),11(a,1)),flip(b)].">
      <j1 rule="para" parents="121 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(92(a,2),121(a,1,1,1))].">
      <j1 rule="para" parents="92 121"/>
    </justification>
  </clause>

  <clause id="143">
    <literal><![CDATA[
      x * y * z != y * u
    ]]></literal>
    <literal><![CDATA[
      x * (x @ y) * z = u
    ]]></literal>
    <justification jstring="[para(13(a,1),1(a,1))].">
      <j1 rule="para" parents="13 1"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      x * y * z * (z @ (x @ y)) = y * x * z * (x @ y)
    ]]></literal>
    <justification jstring="[para(4(a,1),13(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="4 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(73,a,121,a),flip(a)].">
      <j1 rule="hyper" parents="73 121"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(73,a,65,a)].">
      <j1 rule="hyper" parents="73 65"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(167)].">
      <j1 rule="new_symbol" parents="167"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(166),rewrite(171(1),171(3))].">
      <j1 rule="back_rewrite" parents="166"/>
      <j2 rule="rewrite" parents="171 171"/>
    </justification>
  </clause>

  <clause id="181">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_rewrite(141),rewrite(171(1),175(2),175(2))].">
      <j1 rule="back_rewrite" parents="141"/>
      <j2 rule="rewrite" parents="171 175 175"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      x * (x @ y) != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(140),rewrite(171(5),175(6))].">
      <j1 rule="back_rewrite" parents="140"/>
      <j2 rule="rewrite" parents="171 175"/>
    </justification>
  </clause>

  <clause id="184">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(137),rewrite(171(3),175(4),175(4))].">
      <j1 rule="back_rewrite" parents="137"/>
      <j2 rule="rewrite" parents="171 175 175"/>
    </justification>
  </clause>

  <clause id="188">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_rewrite(131),rewrite(171(1),175(2),171(4),175(5))].">
      <j1 rule="back_rewrite" parents="131"/>
      <j2 rule="rewrite" parents="171 175 171 175"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(122),rewrite(171(1),175(2),175(2))].">
      <j1 rule="back_rewrite" parents="122"/>
      <j2 rule="rewrite" parents="171 175 175"/>
    </justification>
  </clause>

  <clause id="194">
    <literal><![CDATA[
      x * y * z != x * y
    ]]></literal>
    <literal><![CDATA[
      c_0 = z
    ]]></literal>
    <justification jstring="[back_rewrite(100),rewrite(171(5),175(6))].">
      <j1 rule="back_rewrite" parents="100"/>
      <j2 rule="rewrite" parents="171 175"/>
    </justification>
  </clause>

  <clause id="196">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_rewrite(188),rewrite(191(5))].">
      <j1 rule="back_rewrite" parents="188"/>
      <j2 rule="rewrite" parents="191"/>
    </justification>
  </clause>

  <clause id="197">
    <literal><![CDATA[
      x * y * (y @ y * x) = y * x
    ]]></literal>
    <justification jstring="[hyper(1,a,14,a)].">
      <j1 rule="hyper" parents="1 14"/>
    </justification>
  </clause>

  <clause id="246">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(184,a,196,a),flip(a)].">
      <j1 rule="hyper" parents="184 196"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="299">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      x @ y = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),194(a,1)),flip(b)].">
      <j1 rule="para" parents="4 194"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="322">
    <literal><![CDATA[
      x * (y * x @ y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(10,a,81,a),flip(a)].">
      <j1 rule="hyper" parents="10 81"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="352">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(10,a,197,a),flip(a)].">
      <j1 rule="hyper" parents="10 197"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="372">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      z = y
    ]]></literal>
    <justification jstring="[para(181(a,1),43(a,1,2)),rewrite(191(3),246(5),191(5))].">
      <j1 rule="para" parents="181 43"/>
      <j2 rule="rewrite" parents="191 246 191"/>
    </justification>
  </clause>

  <clause id="385">
    <literal><![CDATA[
      x * y @ x = y @ x
    ]]></literal>
    <justification jstring="[hyper(372,a,322,a),flip(a)].">
      <j1 rule="hyper" parents="372 322"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="396">
    <literal><![CDATA[
      (x @ y) * y * x = x * y
    ]]></literal>
    <justification jstring="[hyper(45,a,13,a)].">
      <j1 rule="hyper" parents="45 13"/>
    </justification>
  </clause>

  <clause id="397">
    <literal><![CDATA[
      (x @ y) * z * y * x = z * x * y
    ]]></literal>
    <justification jstring="[hyper(45,a,13,a(flip)),flip(a)].">
      <j1 rule="hyper" parents="45 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="401">
    <literal><![CDATA[
      x * (x @ y) @ y = x * y @ y
    ]]></literal>
    <justification jstring="[para(4(a,1),385(a,1,1)),flip(a)].">
      <j1 rule="para" parents="4 385"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="420">
    <literal><![CDATA[
      ((A @ B) @ C) * (F @ G) * D != D * (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(396(a,1),23(a,2))].">
      <j1 rule="para" parents="396 23"/>
    </justification>
  </clause>

  <clause id="459">
    <literal><![CDATA[
      x * y @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(396(a,1),299(a,2)),rewrite(3(3),4(3)),xx(a)].">
      <j1 rule="para" parents="396 299"/>
      <j2 rule="rewrite" parents="3 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="471">
    <literal><![CDATA[
      x * y @ (x @ y) = c_0
    ]]></literal>
    <justification jstring="[para(396(a,1),385(a,1,1)),rewrite(459(6))].">
      <j1 rule="para" parents="396 385"/>
      <j2 rule="rewrite" parents="459"/>
    </justification>
  </clause>

  <clause id="493">
    <literal><![CDATA[
      x * y * z * u != v * u * z
    ]]></literal>
    <literal><![CDATA[
      x * y * (z @ u) = v
    ]]></literal>
    <justification jstring="[para(396(a,1),52(a,1,2,2))].">
      <j1 rule="para" parents="396 52"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      ((A @ B) @ C) * (F @ G) * D * x != D * (F @ G) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(52,b,420,a),rewrite(3(18))].">
      <j1 rule="ur" parents="52 420"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      x * y @ (y @ x * (x @ y)) = c_0
    ]]></literal>
    <justification jstring="[para(4(a,1),471(a,1,1))].">
      <j1 rule="para" parents="4 471"/>
    </justification>
  </clause>

  <clause id="528">
    <literal><![CDATA[
      (x @ y * x) * y * x = x * y
    ]]></literal>
    <justification jstring="[hyper(53,a,396,a)].">
      <j1 rule="hyper" parents="53 396"/>
    </justification>
  </clause>

  <clause id="716">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(372,a,352,a),flip(a)].">
      <j1 rule="hyper" parents="372 352"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="734">
    <literal><![CDATA[
      x * (x @ y * x) = (x @ y) * x
    ]]></literal>
    <justification jstring="[hyper(82,a,13,a)].">
      <j1 rule="hyper" parents="82 13"/>
    </justification>
  </clause>

  <clause id="745">
    <literal><![CDATA[
      x @ y * (y @ x) = x @ y * x
    ]]></literal>
    <justification jstring="[para(4(a,1),716(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 716"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="772">
    <literal><![CDATA[
      x * y @ (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(510),rewrite(745(4))].">
      <j1 rule="back_rewrite" parents="510"/>
      <j2 rule="rewrite" parents="745"/>
    </justification>
  </clause>

  <clause id="849">
    <literal><![CDATA[
      x * (y @ x) * y = y * x
    ]]></literal>
    <justification jstring="[para(772(a,1),4(a,1,2,2)),rewrite(191(5),528(4),3(5),734(4)),flip(a)].">
      <j1 rule="para" parents="772 4"/>
      <j2 rule="rewrite" parents="191 528 3 734"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="908">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(10,a,849,a),flip(a)].">
      <j1 rule="hyper" parents="10 849"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="953">
    <literal><![CDATA[
      x * (x @ y * x) = x * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(734),rewrite(908(5))].">
      <j1 rule="back_rewrite" parents="734"/>
      <j2 rule="rewrite" parents="908"/>
    </justification>
  </clause>

  <clause id="961">
    <literal><![CDATA[
      x * (x @ y) * (y @ x) = x
    ]]></literal>
    <justification jstring="[hyper(143,a,4,a)].">
      <j1 rule="hyper" parents="143 4"/>
    </justification>
  </clause>

  <clause id="1148">
    <literal><![CDATA[
      (x @ y) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(184,a,961,a),flip(a)].">
      <j1 rule="hyper" parents="184 961"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1149">
    <literal><![CDATA[
      (x @ y) * (y @ x) * x = (y @ x) * x * (x @ y)
    ]]></literal>
    <justification jstring="[para(961(a,1),5(a,1,2,2)),rewrite(908(3),1148(9),191(8)),flip(a)].">
      <j1 rule="para" parents="961 5"/>
      <j2 rule="rewrite" parents="908 1148 191"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1185">
    <literal><![CDATA[
      (x @ y) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(1148(a,1),3(a,1,1)),rewrite(181(2)),flip(a)].">
      <j1 rule="para" parents="1148 3"/>
      <j2 rule="rewrite" parents="181"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1233">
    <literal><![CDATA[
      ((A @ B) @ C) * (F @ G) * D * (G @ F) != D
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(1148(a,1),498(a,2,2)),rewrite(191(18))].">
      <j1 rule="para" parents="1148 498"/>
      <j2 rule="rewrite" parents="191"/>
    </justification>
  </clause>

  <clause id="1248">
    <literal><![CDATA[
      (x @ y) * y * (y @ x) = y
    ]]></literal>
    <justification jstring="[back_rewrite(1149),rewrite(1185(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="1149"/>
      <j2 rule="rewrite" parents="1185"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1382">
    <literal><![CDATA[
      (x @ y) * z @ (y @ x) = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(1185(a,1),385(a,1,1)),flip(a)].">
      <j1 rule="para" parents="1185 385"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1403">
    <literal><![CDATA[
      (x @ y) @ (y @ x) * z = (x @ y) @ z
    ]]></literal>
    <justification jstring="[para(1185(a,1),716(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1185 716"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1517">
    <literal><![CDATA[
      x * (x @ y) @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(1248(a,1),299(a,2)),rewrite(3(4),1148(3),191(2)),xx(a)].">
      <j1 rule="para" parents="1248 299"/>
      <j2 rule="rewrite" parents="3 1148 191"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1526">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(1248(a,1),385(a,1,1)),rewrite(1517(6))].">
      <j1 rule="para" parents="1248 385"/>
      <j2 rule="rewrite" parents="1517"/>
    </justification>
  </clause>

  <clause id="1835">
    <literal><![CDATA[
      x * (x @ y) @ y * (y @ x) = y * x @ y * (y @ x)
    ]]></literal>
    <justification jstring="[para(4(a,1),401(a,2,1)),rewrite(745(3),953(3))].">
      <j1 rule="para" parents="4 401"/>
      <j2 rule="rewrite" parents="745 953"/>
    </justification>
  </clause>

  <clause id="1843">
    <literal><![CDATA[
      x * (x @ y) @ x * y = x * x * y @ x * y
    ]]></literal>
    <justification jstring="[para(716(a,1),401(a,1,1,2))].">
      <j1 rule="para" parents="716 401"/>
    </justification>
  </clause>

  <clause id="2149">
    <literal><![CDATA[
      x * y @ x * (x @ y) = x * y @ x * x * y
    ]]></literal>
    <justification jstring="[para(716(a,1),745(a,1,2,2))].">
      <j1 rule="para" parents="716 745"/>
    </justification>
  </clause>

  <clause id="2169">
    <literal><![CDATA[
      x * x * y @ x * y = y * x @ y * y * x
    ]]></literal>
    <justification jstring="[para(745(a,1),745(a,1,2,2)),rewrite(953(5),1835(5),2149(4),4(9),1843(8)),flip(a)].">
      <j1 rule="para" parents="745 745"/>
      <j2 rule="rewrite" parents="953 1835 2149 4 1843"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2298">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[hyper(372,a,953,a),flip(a)].">
      <j1 rule="hyper" parents="372 953"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2317">
    <literal><![CDATA[
      x * x * y @ x * y = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(2169),rewrite(2298(8),385(6))].">
      <j1 rule="back_rewrite" parents="2169"/>
      <j2 rule="rewrite" parents="2298 385"/>
    </justification>
  </clause>

  <clause id="2320">
    <literal><![CDATA[
      x * y @ x * (x @ y) = y @ x
    ]]></literal>
    <justification jstring="[back_rewrite(2149),rewrite(2298(8),385(6))].">
      <j1 rule="back_rewrite" parents="2149"/>
      <j2 rule="rewrite" parents="2298 385"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x * (x @ y) @ x * y = x @ y
    ]]></literal>
    <justification jstring="[back_rewrite(1843),rewrite(2317(8))].">
      <j1 rule="back_rewrite" parents="1843"/>
      <j2 rule="rewrite" parents="2317"/>
    </justification>
  </clause>

  <clause id="2343">
    <literal><![CDATA[
      x @ y * z * x = x @ y * z
    ]]></literal>
    <justification jstring="[para(3(a,1),2298(a,1,2))].">
      <j1 rule="para" parents="3 2298"/>
    </justification>
  </clause>

  <clause id="2344">
    <literal><![CDATA[
      x * y @ y = x @ y
    ]]></literal>
    <justification jstring="[para(4(a,1),2298(a,1,2)),rewrite(2340(4),401(4)),flip(a)].">
      <j1 rule="para" parents="4 2298"/>
      <j2 rule="rewrite" parents="2340 401"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2360">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(1185(a,1),2298(a,1,2)),rewrite(2344(3),1382(6))].">
      <j1 rule="para" parents="1185 2298"/>
      <j2 rule="rewrite" parents="2344 1382"/>
    </justification>
  </clause>

  <clause id="2551">
    <literal><![CDATA[
      (C @ (B @ A)) * (F @ G) * D * (G @ F) != D
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(1233),rewrite(2360(5))].">
      <j1 rule="back_rewrite" parents="1233"/>
      <j2 rule="rewrite" parents="2360"/>
    </justification>
  </clause>

  <clause id="2844">
    <literal><![CDATA[
      (x @ y) @ ((y @ x) @ z) = c_0
    ]]></literal>
    <justification jstring="[para(2360(a,2),1526(a,1,2))].">
      <j1 rule="para" parents="2360 1526"/>
    </justification>
  </clause>

  <clause id="2919">
    <literal><![CDATA[
      x * ((y @ z) @ x) = (y @ z) * x * (z @ y)
    ]]></literal>
    <justification jstring="[para(2844(a,1),148(a,1,2,2,2)),rewrite(191(4),1185(10)),flip(a)].">
      <j1 rule="para" parents="2844 148"/>
      <j2 rule="rewrite" parents="191 1185"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3009">
    <literal><![CDATA[
      x * y @ y * y * x = x @ y
    ]]></literal>
    <justification jstring="[para(2320(a,1),2340(a,1,1,2)),rewrite(3(3),4(3),3(5),4(4),2320(8))].">
      <j1 rule="para" parents="2320 2340"/>
      <j2 rule="rewrite" parents="3 4 3 4 2320"/>
    </justification>
  </clause>

  <clause id="3531">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x) * z * (y @ x) * z * (x @ y)
    ]]></literal>
    <justification jstring="[para(1185(a,1),3009(a,1,1)),rewrite(3(6),3(7),1403(12)),flip(a)].">
      <j1 rule="para" parents="1185 3009"/>
      <j2 rule="rewrite" parents="3 3 1403"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4670">
    <literal><![CDATA[
      x * y * z @ (z @ y) = c_0
    ]]></literal>
    <justification jstring="[para(397(a,1),299(a,2)),rewrite(3(4),3(3),4(3)),xx(a)].">
      <j1 rule="para" parents="397 299"/>
      <j2 rule="rewrite" parents="3 3 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="5118">
    <literal><![CDATA[
      x * y @ ((z @ u) @ y) = c_0
    ]]></literal>
    <justification jstring="[para(1185(a,1),4670(a,1,1,2)),rewrite(2360(5,R),716(5))].">
      <j1 rule="para" parents="1185 4670"/>
      <j2 rule="rewrite" parents="2360 716"/>
    </justification>
  </clause>

  <clause id="5337">
    <literal><![CDATA[
      x * (y @ z) * u * (z @ y) != x * u
    ]]></literal>
    <literal><![CDATA[
      (y @ z) @ u = c_0
    ]]></literal>
    <justification jstring="[para(5118(a,1),182(a,1,2)),rewrite(191(3),3(5),2919(4)),flip(a),flip(b)].">
      <j1 rule="para" parents="5118 182"/>
      <j2 rule="rewrite" parents="191 3 2919"/>
      <j2 rule="flip"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10297">
    <literal><![CDATA[
      (x @ y) * z * (y @ x) = z
    ]]></literal>
    <justification jstring="[hyper(493,a,397,a)].">
      <j1 rule="hyper" parents="493 397"/>
    </justification>
  </clause>

  <clause id="10361">
    <literal><![CDATA[
      (x @ y) @ z = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(5337),rewrite(10297(4)),xx(a)].">
      <j1 rule="back_rewrite" parents="5337"/>
      <j2 rule="rewrite" parents="10297"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="10401">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(3531),rewrite(10361(2),10297(6),2343(5),2298(4)),flip(a)].">
      <j1 rule="back_rewrite" parents="3531"/>
      <j2 rule="rewrite" parents="10361 10297 2343 2298"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10404">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(2551),rewrite(10401(5),10297(10),181(3)),xx(a)].">
      <j1 rule="back_rewrite" parents="2551"/>
      <j2 rule="rewrite" parents="10401 10297 181"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
