<?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>x3-ring.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3285 was started by mccune on cleo.thornwood,
Wed Nov 22 11:21:31 2006
The command was "/home/mccune/bin/prover9 -f x3-ring.in".
]]></heading>

<proof number="1" length="78" max_count="20">

<comments><![CDATA[
% Proof 1 at 4.33 (+ 0.04) seconds: commutativity.
% Length of proof is 78.
% Level of proof is 26.
% Maximum clause weight is 20.
% Given clauses 203.
]]></comments>

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

  <clause id="2" type="assumption">
    <literal><![CDATA[
      x + 0 = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      x + -x = 0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

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

  <clause id="7">
    <literal><![CDATA[
      (x * y) + (x * z) = x * (y + z)
    ]]></literal>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      (x + y) * z = (x * z) + (y * z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      (x * y) + (z * y) = (x + z) * y
    ]]></literal>
    <justification jstring="[copy(8),flip(a)].">
      <j1 rule="copy" parents="8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      (x * y) * z = x * (y * z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x * (x * x) = x
    ]]></literal>
    <attribute><![CDATA[
      label("hypothesis x^3=x")
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12" type="deny">
    <literal><![CDATA[
      c2 * c1 != c1 * c2
    ]]></literal>
    <attribute><![CDATA[
      label(commutativity)
    ]]></attribute>
    <attribute><![CDATA[
      answer(commutativity)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x + (-x + y) = 0 + y
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      -x + (y + x) = y
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,2,2)),rewrite(5(3),2(5))].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="rewrite" parents="5 2"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      0 + x = x
    ]]></literal>
    <justification jstring="[para(5(a,1),2(a,1))].">
      <j1 rule="para" parents="5 2"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(5(a,1),4(a,1,1)),rewrite(4(2))].">
      <j1 rule="para" parents="5 4"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x + (-x + y) = y
    ]]></literal>
    <justification jstring="[back_rewrite(16),rewrite(18(5))].">
      <j1 rule="back_rewrite" parents="16"/>
      <j2 rule="rewrite" parents="18"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      (x * (y * z)) + (u * z) = ((x * y) + u) * z
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      (x * y) + (z * (u * y)) = (x + (z * u)) * y
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,2))].">
      <j1 rule="para" parents="10 9"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x * ((x * x) + y) = x + (x * y)
    ]]></literal>
    <justification jstring="[para(11(a,1),7(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * (x * (x * y)) = x * y
    ]]></literal>
    <justification jstring="[para(11(a,1),10(a,1,1)),rewrite(10(3)),flip(a)].">
      <j1 rule="para" parents="11 10"/>
      <j2 rule="rewrite" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x * (y * (x * (y * (x * y)))) = x * y
    ]]></literal>
    <justification jstring="[para(11(a,1),10(a,1)),rewrite(10(4)),flip(a)].">
      <j1 rule="para" parents="11 10"/>
      <j2 rule="rewrite" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      --x = x
    ]]></literal>
    <justification jstring="[para(3(a,1),17(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="3 17"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      -x + (y + (x + z)) = y + z
    ]]></literal>
    <justification jstring="[para(17(a,1),4(a,1,1)),rewrite(4(4)),flip(a)].">
      <j1 rule="para" parents="17 4"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      -(x * y) + (x * (z + y)) = x * z
    ]]></literal>
    <justification jstring="[para(7(a,1),17(a,1,2))].">
      <j1 rule="para" parents="7 17"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x + -(x + y) = -y
    ]]></literal>
    <justification jstring="[para(17(a,1),17(a,1,2)),rewrite(5(3))].">
      <j1 rule="para" parents="17 17"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (x * y) + (z + (x * u)) = z + (x * (y + u))
    ]]></literal>
    <justification jstring="[para(7(a,1),19(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      (x * y) + -(x * (y + z)) = -(x * z)
    ]]></literal>
    <justification jstring="[para(7(a,1),41(a,1,2,1))].">
      <j1 rule="para" parents="7 41"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      -(x + y) = -y + -x
    ]]></literal>
    <justification jstring="[para(17(a,1),41(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="17 41"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      ((x * x) + y) * x = x + (y * x)
    ]]></literal>
    <justification jstring="[para(11(a,1),24(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="173">
    <literal><![CDATA[
      ((x * x) + y) * (x * z) = (x + (y * x)) * z
    ]]></literal>
    <justification jstring="[para(29(a,1),24(a,1,1)),rewrite(25(4)),flip(a)].">
      <j1 rule="para" parents="29 24"/>
      <j2 rule="rewrite" parents="25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      x + (x * 0) = x
    ]]></literal>
    <justification jstring="[para(2(a,1),26(a,1,2)),rewrite(11(2)),flip(a)].">
      <j1 rule="para" parents="2 26"/>
      <j2 rule="rewrite" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      x + (x * -(x * x)) = x * 0
    ]]></literal>
    <justification jstring="[para(3(a,1),26(a,1,2)),flip(a)].">
      <j1 rule="para" parents="3 26"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      x * (y + (0 * y)) = x * y
    ]]></literal>
    <justification jstring="[para(176(a,1),9(a,2,1)),rewrite(10(4),7(5))].">
      <j1 rule="para" parents="176 9"/>
      <j2 rule="rewrite" parents="10 7"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      0 * 0 = 0
    ]]></literal>
    <justification jstring="[para(176(a,1),18(a,1)),flip(a)].">
      <j1 rule="para" parents="176 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="214">
    <literal><![CDATA[
      x + (y * 0) = x
    ]]></literal>
    <justification jstring="[para(176(a,1),36(a,1,2,2)),rewrite(17(3)),flip(a)].">
      <j1 rule="para" parents="176 36"/>
      <j2 rule="rewrite" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="216">
    <literal><![CDATA[
      x * 0 = 0
    ]]></literal>
    <justification jstring="[para(208(a,1),9(a,1,1)),rewrite(214(4),18(3)),flip(a)].">
      <j1 rule="para" parents="208 9"/>
      <j2 rule="rewrite" parents="214 18"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="218">
    <literal><![CDATA[
      x + (x * -(x * x)) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(177),rewrite(216(6))].">
      <j1 rule="back_rewrite" parents="177"/>
      <j2 rule="rewrite" parents="216"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      (x + (y * z)) * (z * z) = ((x * z) + y) * z
    ]]></literal>
    <justification jstring="[para(11(a,1),25(a,1,2,2)),rewrite(24(4)),flip(a)].">
      <j1 rule="para" parents="11 25"/>
      <j2 rule="rewrite" parents="24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="251">
    <literal><![CDATA[
      x * (y + (-(x * x) * y)) = 0 * y
    ]]></literal>
    <justification jstring="[para(218(a,1),9(a,2,1)),rewrite(10(5),7(6))].">
      <j1 rule="para" parents="218 9"/>
      <j2 rule="rewrite" parents="10 7"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      -(x * -(x * x)) = x
    ]]></literal>
    <justification jstring="[para(218(a,1),17(a,1,2)),rewrite(5(6),18(6))].">
      <j1 rule="para" parents="218 17"/>
      <j2 rule="rewrite" parents="5 18"/>
    </justification>
  </clause>

  <clause id="266">
    <literal><![CDATA[
      x * -(x * x) = -x
    ]]></literal>
    <justification jstring="[para(253(a,1),35(a,1,1)),flip(a)].">
      <j1 rule="para" parents="253 35"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="279">
    <literal><![CDATA[
      x * (x * -x) = -x
    ]]></literal>
    <justification jstring="[para(266(a,1),29(a,1,2,2)),rewrite(266(6))].">
      <j1 rule="para" parents="266 29"/>
      <j2 rule="rewrite" parents="266"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      x * (x * (-x * y)) = -x * y
    ]]></literal>
    <justification jstring="[para(279(a,1),10(a,1,1)),rewrite(10(5)),flip(a)].">
      <j1 rule="para" parents="279 10"/>
      <j2 rule="rewrite" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="390">
    <literal><![CDATA[
      x * ((y + (0 * y)) * z) = x * (y * z)
    ]]></literal>
    <justification jstring="[para(207(a,1),10(a,1,1)),rewrite(10(2)),flip(a)].">
      <j1 rule="para" parents="207 10"/>
      <j2 rule="rewrite" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="391">
    <literal><![CDATA[
      x + (0 * x) = x
    ]]></literal>
    <justification jstring="[para(207(a,1),11(a,1,2)),rewrite(390(8),222(5),5(3),18(3),10(2),11(2)),flip(a)].">
      <j1 rule="para" parents="207 11"/>
      <j2 rule="rewrite" parents="390 222 5 18 10 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="395">
    <literal><![CDATA[
      0 * -x = 0
    ]]></literal>
    <justification jstring="[para(391(a,1),20(a,1,2)),rewrite(3(2)),flip(a)].">
      <j1 rule="para" parents="391 20"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="410">
    <literal><![CDATA[
      0 * x = 0
    ]]></literal>
    <justification jstring="[para(35(a,1),395(a,1,2))].">
      <j1 rule="para" parents="35 395"/>
    </justification>
  </clause>

  <clause id="412">
    <literal><![CDATA[
      x * (y + (-(x * x) * y)) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(251),rewrite(410(7))].">
      <j1 rule="back_rewrite" parents="251"/>
      <j2 rule="rewrite" parents="410"/>
    </justification>
  </clause>

  <clause id="431">
    <literal><![CDATA[
      x * ((y + (-(x * x) * y)) * z) = 0
    ]]></literal>
    <justification jstring="[para(412(a,1),10(a,1,1)),rewrite(410(2)),flip(a)].">
      <j1 rule="para" parents="412 10"/>
      <j2 rule="rewrite" parents="410"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="492">
    <literal><![CDATA[
      -(x * y) = x * -y
    ]]></literal>
    <justification jstring="[para(17(a,1),38(a,1,2,2)),rewrite(5(5),60(5))].">
      <j1 rule="para" parents="17 38"/>
      <j2 rule="rewrite" parents="5 60"/>
    </justification>
  </clause>

  <clause id="501">
    <literal><![CDATA[
      -x * -y = x * y
    ]]></literal>
    <justification jstring="[para(412(a,1),38(a,1,2)),rewrite(492(2),10(3),305(4),492(3),5(5),18(5))].">
      <j1 rule="para" parents="412 38"/>
      <j2 rule="rewrite" parents="492 10 305 492 5 18"/>
    </justification>
  </clause>

  <clause id="529">
    <literal><![CDATA[
      x * ((y + (x * (-x * y))) * z) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(431),rewrite(492(2),10(3))].">
      <j1 rule="back_rewrite" parents="431"/>
      <j2 rule="rewrite" parents="492 10"/>
    </justification>
  </clause>

  <clause id="569">
    <literal><![CDATA[
      (x * -y) + (z * y) = (x + -z) * -y
    ]]></literal>
    <justification jstring="[para(501(a,1),9(a,1,2))].">
      <j1 rule="para" parents="501 9"/>
    </justification>
  </clause>

  <clause id="572">
    <literal><![CDATA[
      -x * y = x * -y
    ]]></literal>
    <justification jstring="[para(35(a,1),501(a,1,1)),flip(a)].">
      <j1 rule="para" parents="35 501"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="579">
    <literal><![CDATA[
      (x * y) + (z * (u * -y)) = (-x + (z * u)) * -y
    ]]></literal>
    <justification jstring="[para(501(a,1),25(a,1,1))].">
      <j1 rule="para" parents="501 25"/>
    </justification>
  </clause>

  <clause id="590">
    <literal><![CDATA[
      x * ((y + (x * (x * -y))) * z) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(529),rewrite(572(2))].">
      <j1 rule="back_rewrite" parents="529"/>
      <j2 rule="rewrite" parents="572"/>
    </justification>
  </clause>

  <clause id="608">
    <literal><![CDATA[
      (-x + y) * z = (x + -y) * -z
    ]]></literal>
    <justification jstring="[para(572(a,1),9(a,1,1)),rewrite(569(4)),flip(a)].">
      <j1 rule="para" parents="572 9"/>
      <j2 rule="rewrite" parents="569"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="612">
    <literal><![CDATA[
      (x * y) + (z * (u * -y)) = (x + (z * -u)) * y
    ]]></literal>
    <justification jstring="[back_rewrite(579),rewrite(608(10),492(7),35(10))].">
      <j1 rule="back_rewrite" parents="579"/>
      <j2 rule="rewrite" parents="608 492 35"/>
    </justification>
  </clause>

  <clause id="652">
    <literal><![CDATA[
      x + ((y + (x * z)) * x) = (y + (x * (x + z))) * x
    ]]></literal>
    <justification jstring="[para(44(a,1),157(a,1,1)),flip(a)].">
      <j1 rule="para" parents="44 157"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="683">
    <literal><![CDATA[
      (x + (y * (y * -x))) * y = 0
    ]]></literal>
    <justification jstring="[para(590(a,1),30(a,1,2,2,2)),rewrite(216(10),216(6),216(6)),flip(a)].">
      <j1 rule="para" parents="590 30"/>
      <j2 rule="rewrite" parents="216 216 216"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="692">
    <literal><![CDATA[
      (x * y) + (y * (y * (x * -y))) = 0
    ]]></literal>
    <justification jstring="[para(683(a,1),9(a,2)),rewrite(10(5),10(4),572(3))].">
      <j1 rule="para" parents="683 9"/>
      <j2 rule="rewrite" parents="10 10 572"/>
    </justification>
  </clause>

  <clause id="1405">
    <literal><![CDATA[
      x * (x * (y * x)) = y * x
    ]]></literal>
    <justification jstring="[para(692(a,1),17(a,1,2)),rewrite(492(5),492(4),492(3),35(2),5(5),18(5))].">
      <j1 rule="para" parents="692 17"/>
      <j2 rule="rewrite" parents="492 492 492 35 5 18"/>
    </justification>
  </clause>

  <clause id="1437">
    <literal><![CDATA[
      x * (x * (y * (x * z))) = y * (x * z)
    ]]></literal>
    <justification jstring="[para(1405(a,1),10(a,1,1)),rewrite(10(2),10(5),10(4)),flip(a)].">
      <j1 rule="para" parents="1405 10"/>
      <j2 rule="rewrite" parents="10 10 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1453">
    <literal><![CDATA[
      x * (y * (x * (y * (y * x)))) = y * x
    ]]></literal>
    <justification jstring="[para(30(a,1),1405(a,1,2,2)),rewrite(10(10),10(9),10(8),10(7),30(9),10(7),10(6),10(5),10(4),1405(3),30(10))].">
      <j1 rule="para" parents="30 1405"/>
      <j2 rule="rewrite" parents="10 10 10 10 30 10 10 10 10 1405 30"/>
    </justification>
  </clause>

  <clause id="1455">
    <literal><![CDATA[
      x * (x * (y * -x)) = y * -x
    ]]></literal>
    <justification jstring="[para(1405(a,1),492(a,1,1)),rewrite(492(2),492(5),492(4)),flip(a)].">
      <j1 rule="para" parents="1405 492"/>
      <j2 rule="rewrite" parents="492 492 492"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1456">
    <literal><![CDATA[
      x * (y * (z + (x * (x * -z)))) = 0
    ]]></literal>
    <justification jstring="[para(1405(a,1),590(a,1,2))].">
      <j1 rule="para" parents="1405 590"/>
    </justification>
  </clause>

  <clause id="1512">
    <literal><![CDATA[
      x * (y + (x * (z * -x))) = (x * y) + (z * -x)
    ]]></literal>
    <justification jstring="[para(1455(a,1),7(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1455 7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1771">
    <literal><![CDATA[
      x * (y + (y * (x * (x * (y * -y))))) = 0
    ]]></literal>
    <justification jstring="[para(26(a,1),1456(a,1,2)),rewrite(492(2))].">
      <j1 rule="para" parents="26 1456"/>
      <j2 rule="rewrite" parents="492"/>
    </justification>
  </clause>

  <clause id="4628">
    <literal><![CDATA[
      x * (y * (x * (y * x))) = y * (y * x)
    ]]></literal>
    <justification jstring="[para(30(a,1),1437(a,1,2)),flip(a)].">
      <j1 rule="para" parents="30 1437"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5170">
    <literal><![CDATA[
      x * (y * (x * (x * y))) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(1405(a,1),1453(a,1,2,2,2)),rewrite(10(4),4628(5),10(4))].">
      <j1 rule="para" parents="1405 1453"/>
      <j2 rule="rewrite" parents="10 4628 10"/>
    </justification>
  </clause>

  <clause id="7509">
    <literal><![CDATA[
      x * (y * (x * (x * (y * z)))) = y * (x * (y * z))
    ]]></literal>
    <justification jstring="[para(5170(a,1),10(a,1,1)),rewrite(10(3),10(2),10(7),10(6),10(5)),flip(a)].">
      <j1 rule="para" parents="5170 10"/>
      <j2 rule="rewrite" parents="10 10 10 10 10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10288">
    <literal><![CDATA[
      (x * y) + (y * (x * (y * -y))) = 0
    ]]></literal>
    <justification jstring="[para(1771(a,1),7(a,2)),rewrite(7509(7))].">
      <j1 rule="para" parents="1771 7"/>
      <j2 rule="rewrite" parents="7509"/>
    </justification>
  </clause>

  <clause id="10372">
    <literal><![CDATA[
      (x + (y * (x * -y))) * y = 0
    ]]></literal>
    <justification jstring="[para(157(a,1),10288(a,1,1)),rewrite(173(7),4(8),612(7),62(3),492(2),5(4),652(8),20(5))].">
      <j1 rule="para" parents="157 10288"/>
      <j2 rule="rewrite" parents="173 4 612 62 492 5 652 20"/>
    </justification>
  </clause>

  <clause id="11568">
    <literal><![CDATA[
      (x + (y * (x * -y))) * (y * z) = 0
    ]]></literal>
    <justification jstring="[para(10372(a,1),10(a,1,1)),rewrite(410(2)),flip(a)].">
      <j1 rule="para" parents="10372 10"/>
      <j2 rule="rewrite" parents="410"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13251">
    <literal><![CDATA[
      (x * y) + (y * -x) = 0
    ]]></literal>
    <justification jstring="[para(11568(a,1),1405(a,1,2)),rewrite(216(6),1512(6)),flip(a)].">
      <j1 rule="para" parents="11568 1405"/>
      <j2 rule="rewrite" parents="216 1512"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13324">
    <literal><![CDATA[
      x * y = y * x
    ]]></literal>
    <justification jstring="[para(13251(a,1),17(a,1,2)),rewrite(492(3),35(2),5(3),18(3))].">
      <j1 rule="para" parents="13251 17"/>
      <j2 rule="rewrite" parents="492 35 5 18"/>
    </justification>
  </clause>

  <clause id="13325">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(commutativity)
    ]]></attribute>
    <justification jstring="[resolve(13324,a,12,a)].">
      <j1 rule="resolve" parents="13324 12"/>
    </justification>
  </clause>

</proof>

</proofs>
