<?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 August-2006A, August 2006.
Process 10608 was started by mccune on cleo.thornwood,
Sat Aug 12 20:57:15 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 3.95 (+ 0.03) seconds.
% Length of proof is 78.
% Level of proof is 26.
% Maximum clause weight is 20.
% Given clauses 203.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      x + 0 = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

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

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

  <clause id="6">
    <literal><![CDATA[
      (x * y) + (x * z) = x * (y + z)
    ]]></literal>
    <justification jstring="[copy(5),flip(a)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="8">
    <literal><![CDATA[
      (x * y) + (z * y) = (x + z) * y
    ]]></literal>
    <justification jstring="[copy(7),flip(a)].">
      <j1 rule="copy" parents="7"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

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

  <clause id="11">
    <literal><![CDATA[
      A * B != B * A
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      B * A != A * B
    ]]></literal>
    <justification jstring="[copy(11),flip(a)].">
      <j1 rule="copy" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      x + (-x + y) = 0 + y
    ]]></literal>
    <justification jstring="[para(2(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="2 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="18">
    <literal><![CDATA[
      0 + x = x
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1))].">
      <j1 rule="para" parents="4 1"/>
    </justification>
  </clause>

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

  <clause id="25">
    <literal><![CDATA[
      (x * y) + (z * (u * y)) = (x + (z * u)) * y
    ]]></literal>
    <justification jstring="[para(9(a,1),8(a,1,2))].">
      <j1 rule="para" parents="9 8"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x * ((x * x) + y) = x + (x * y)
    ]]></literal>
    <justification jstring="[para(10(a,1),6(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x * (x * (x * y)) = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1,1)),rewrite(9(3)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="rewrite" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x * (y * (x * (y * (x * y)))) = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),9(a,1)),rewrite(9(4)),flip(a)].">
      <j1 rule="para" parents="10 9"/>
      <j2 rule="rewrite" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="36">
    <literal><![CDATA[
      -x + (y + (x + z)) = y + z
    ]]></literal>
    <justification jstring="[para(17(a,1),3(a,1,1)),rewrite(3(4)),flip(a)].">
      <j1 rule="para" parents="17 3"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      -(x * y) + (x * (z + y)) = x * z
    ]]></literal>
    <justification jstring="[para(6(a,1),17(a,1,2))].">
      <j1 rule="para" parents="6 17"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x + -(x + y) = -y
    ]]></literal>
    <justification jstring="[para(17(a,1),17(a,1,2)),rewrite(4(3))].">
      <j1 rule="para" parents="17 17"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (x * y) + (z + (x * u)) = z + (x * (y + u))
    ]]></literal>
    <justification jstring="[para(6(a,1),19(a,1,2)),flip(a)].">
      <j1 rule="para" parents="6 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      (x * y) + -(x * (y + z)) = -(x * z)
    ]]></literal>
    <justification jstring="[para(6(a,1),41(a,1,2,1))].">
      <j1 rule="para" parents="6 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(10(a,1),24(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 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(1(a,1),26(a,1,2)),rewrite(10(2)),flip(a)].">
      <j1 rule="para" parents="1 26"/>
      <j2 rule="rewrite" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="207">
    <literal><![CDATA[
      x * (y + (0 * y)) = x * y
    ]]></literal>
    <justification jstring="[para(176(a,1),8(a,2,1)),rewrite(9(4),6(5))].">
      <j1 rule="para" parents="176 8"/>
      <j2 rule="rewrite" parents="9 6"/>
    </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),8(a,1,1)),rewrite(214(4),18(3)),flip(a)].">
      <j1 rule="para" parents="208 8"/>
      <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(10(a,1),25(a,1,2,2)),rewrite(24(4)),flip(a)].">
      <j1 rule="para" parents="10 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),8(a,2,1)),rewrite(9(5),6(6))].">
      <j1 rule="para" parents="218 8"/>
      <j2 rule="rewrite" parents="9 6"/>
    </justification>
  </clause>

  <clause id="253">
    <literal><![CDATA[
      -(x * -(x * x)) = x
    ]]></literal>
    <justification jstring="[para(218(a,1),17(a,1,2)),rewrite(4(6),18(6))].">
      <j1 rule="para" parents="218 17"/>
      <j2 rule="rewrite" parents="4 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),9(a,1,1)),rewrite(9(5)),flip(a)].">
      <j1 rule="para" parents="279 9"/>
      <j2 rule="rewrite" parents="9"/>
      <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),9(a,1,1)),rewrite(9(2)),flip(a)].">
      <j1 rule="para" parents="207 9"/>
      <j2 rule="rewrite" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="391">
    <literal><![CDATA[
      x + (0 * x) = x
    ]]></literal>
    <justification jstring="[para(207(a,1),10(a,1,2)),rewrite(390(8),222(5),4(3),18(3),9(2),10(2)),flip(a)].">
      <j1 rule="para" parents="207 10"/>
      <j2 rule="rewrite" parents="390 222 4 18 9 10"/>
      <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(2(2)),flip(a)].">
      <j1 rule="para" parents="391 20"/>
      <j2 rule="rewrite" parents="2"/>
      <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),9(a,1,1)),rewrite(410(2)),flip(a)].">
      <j1 rule="para" parents="412 9"/>
      <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(4(5),60(5))].">
      <j1 rule="para" parents="17 38"/>
      <j2 rule="rewrite" parents="4 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),9(3),305(4),492(3),4(5),18(5))].">
      <j1 rule="para" parents="412 38"/>
      <j2 rule="rewrite" parents="492 9 305 492 4 18"/>
    </justification>
  </clause>

  <clause id="529">
    <literal><![CDATA[
      x * ((y + (x * (-x * y))) * z) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(431),rewrite(492(2),9(3))].">
      <j1 rule="back_rewrite" parents="431"/>
      <j2 rule="rewrite" parents="492 9"/>
    </justification>
  </clause>

  <clause id="569">
    <literal><![CDATA[
      (x * -y) + (z * y) = (x + -z) * -y
    ]]></literal>
    <justification jstring="[para(501(a,1),8(a,1,2))].">
      <j1 rule="para" parents="501 8"/>
    </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),8(a,1,1)),rewrite(569(4)),flip(a)].">
      <j1 rule="para" parents="572 8"/>
      <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),8(a,2)),rewrite(9(5),9(4),572(3))].">
      <j1 rule="para" parents="683 8"/>
      <j2 rule="rewrite" parents="9 9 572"/>
    </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),4(5),18(5))].">
      <j1 rule="para" parents="692 17"/>
      <j2 rule="rewrite" parents="492 492 492 35 4 18"/>
    </justification>
  </clause>

  <clause id="1437">
    <literal><![CDATA[
      x * (x * (y * (x * z))) = y * (x * z)
    ]]></literal>
    <justification jstring="[para(1405(a,1),9(a,1,1)),rewrite(9(2),9(5),9(4)),flip(a)].">
      <j1 rule="para" parents="1405 9"/>
      <j2 rule="rewrite" parents="9 9 9"/>
      <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(9(10),9(9),9(8),9(7),30(9),9(7),9(6),9(5),9(4),1405(3),30(10))].">
      <j1 rule="para" parents="30 1405"/>
      <j2 rule="rewrite" parents="9 9 9 9 30 9 9 9 9 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),6(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1455 6"/>
      <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(9(4),4628(5),9(4))].">
      <j1 rule="para" parents="1405 1453"/>
      <j2 rule="rewrite" parents="9 4628 9"/>
    </justification>
  </clause>

  <clause id="7485">
    <literal><![CDATA[
      x * (y * (x * (x * (y * z)))) = y * (x * (y * z))
    ]]></literal>
    <justification jstring="[para(5170(a,1),9(a,1,1)),rewrite(9(3),9(2),9(7),9(6),9(5)),flip(a)].">
      <j1 rule="para" parents="5170 9"/>
      <j2 rule="rewrite" parents="9 9 9 9 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9669">
    <literal><![CDATA[
      (x * y) + (y * (x * (y * -y))) = 0
    ]]></literal>
    <justification jstring="[para(1771(a,1),6(a,2)),rewrite(7485(7))].">
      <j1 rule="para" parents="1771 6"/>
      <j2 rule="rewrite" parents="7485"/>
    </justification>
  </clause>

  <clause id="9724">
    <literal><![CDATA[
      (x + (y * (x * -y))) * y = 0
    ]]></literal>
    <justification jstring="[para(157(a,1),9669(a,1,1)),rewrite(173(7),3(8),612(7),62(3),492(2),4(4),652(8),20(5))].">
      <j1 rule="para" parents="157 9669"/>
      <j2 rule="rewrite" parents="173 3 612 62 492 4 652 20"/>
    </justification>
  </clause>

  <clause id="10329">
    <literal><![CDATA[
      (x + (y * (x * -y))) * (y * z) = 0
    ]]></literal>
    <justification jstring="[para(9724(a,1),9(a,1,1)),rewrite(410(2)),flip(a)].">
      <j1 rule="para" parents="9724 9"/>
      <j2 rule="rewrite" parents="410"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11129">
    <literal><![CDATA[
      (x * y) + (y * -x) = 0
    ]]></literal>
    <justification jstring="[para(10329(a,1),1405(a,1,2)),rewrite(216(6),1512(6)),flip(a)].">
      <j1 rule="para" parents="10329 1405"/>
      <j2 rule="rewrite" parents="216 1512"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11185">
    <literal><![CDATA[
      x * y = y * x
    ]]></literal>
    <justification jstring="[para(11129(a,1),17(a,1,2)),rewrite(492(3),35(2),4(3),18(3))].">
      <j1 rule="para" parents="11129 17"/>
      <j2 rule="rewrite" parents="492 35 4 18"/>
    </justification>
  </clause>

  <clause id="11186">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(11185,a,12,a)].">
      <j1 rule="resolve" parents="11185 12"/>
    </justification>
  </clause>

</proof>

</proofs>
