<?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 June-2006C, June 2006.
Process 12986 was started by mccune on cleo.thornwood,
Mon Jun 19 16:39:37 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 3.98 (+ 0.06) seconds.
% Length of proof is 78.
% Level of proof is 26.
% Maximum clause weight is 20.
% Given clauses 203.
]]></comments>

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

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

  <clause id="9">
    <literal><![CDATA[
      A * B != B * A
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x + 0 = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x + - x = 0
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (x + y) + z = x + (y + z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      x + y = y + x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="14">
    <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="15">
    <literal><![CDATA[
      (x * y) + (z * y) = (x + z) * y
    ]]></literal>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x * y) * z = x * (y * z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x * (x * x) = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      B * A != A * B
    ]]></literal>
    <justification jstring="[copy(9),flip(a)].">
      <j1 rule="copy" parents="9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x + (- x + y) = 0 + y
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      - x + (y + x) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,2,2)),demod(13(3),10(5))].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="demod" parents="13 10"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      0 + x = x
    ]]></literal>
    <justification jstring="[para(13(a,1),10(a,1))].">
      <j1 rule="para" parents="13 10"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x + (y + z) = y + (x + z)
    ]]></literal>
    <justification jstring="[para(13(a,1),12(a,1,1)),demod(12(2))].">
      <j1 rule="para" parents="13 12"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x + (- x + y) = y
    ]]></literal>
    <justification jstring="[back_demod(22),demod(24(5))].">
      <j1 rule="back_demod" parents="22"/>
      <j2 rule="demod" parents="24"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      (x * (y * z)) + (u * z) = ((x * y) + u) * z
    ]]></literal>
    <justification jstring="[para(16(a,1),15(a,1,1))].">
      <j1 rule="para" parents="16 15"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      (x * y) + (z * (u * y)) = (x + (z * u)) * y
    ]]></literal>
    <justification jstring="[para(16(a,1),15(a,1,2))].">
      <j1 rule="para" parents="16 15"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x * ((x * x) + y) = x + (x * y)
    ]]></literal>
    <justification jstring="[para(17(a,1),14(a,1,1)),flip(a)].">
      <j1 rule="para" parents="17 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x * (x * (x * y)) = x * y
    ]]></literal>
    <justification jstring="[para(17(a,1),16(a,1,1)),demod(16(3)),flip(a)].">
      <j1 rule="para" parents="17 16"/>
      <j2 rule="demod" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x * (y * (x * (y * (x * y)))) = x * y
    ]]></literal>
    <justification jstring="[para(17(a,1),16(a,1)),demod(16(4)),flip(a)].">
      <j1 rule="para" parents="17 16"/>
      <j2 rule="demod" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      - - x = x
    ]]></literal>
    <justification jstring="[para(11(a,1),23(a,1,2)),demod(10(4))].">
      <j1 rule="para" parents="11 23"/>
      <j2 rule="demod" parents="10"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      - x + (y + (x + z)) = y + z
    ]]></literal>
    <justification jstring="[para(23(a,1),12(a,1,1)),demod(12(4)),flip(a)].">
      <j1 rule="para" parents="23 12"/>
      <j2 rule="demod" parents="12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      - (x * y) + (x * (z + y)) = x * z
    ]]></literal>
    <justification jstring="[para(14(a,1),23(a,1,2))].">
      <j1 rule="para" parents="14 23"/>
    </justification>
  </clause>

  <clause id="47">
    <literal><![CDATA[
      x + - (x + y) = - y
    ]]></literal>
    <justification jstring="[para(23(a,1),23(a,1,2)),demod(13(3))].">
      <j1 rule="para" parents="23 23"/>
      <j2 rule="demod" parents="13"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      (x * y) + (z + (x * u)) = z + (x * (y + u))
    ]]></literal>
    <justification jstring="[para(14(a,1),25(a,1,2)),flip(a)].">
      <j1 rule="para" parents="14 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="66">
    <literal><![CDATA[
      (x * y) + - (x * (y + z)) = - (x * z)
    ]]></literal>
    <justification jstring="[para(14(a,1),47(a,1,2,1))].">
      <j1 rule="para" parents="14 47"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      - (x + y) = - y + - x
    ]]></literal>
    <justification jstring="[para(23(a,1),47(a,1,2,1)),flip(a)].">
      <j1 rule="para" parents="23 47"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      ((x * x) + y) * x = x + (y * x)
    ]]></literal>
    <justification jstring="[para(17(a,1),30(a,1,1)),flip(a)].">
      <j1 rule="para" parents="17 30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="179">
    <literal><![CDATA[
      ((x * x) + y) * (x * z) = (x + (y * x)) * z
    ]]></literal>
    <justification jstring="[para(35(a,1),30(a,1,1)),demod(31(4)),flip(a)].">
      <j1 rule="para" parents="35 30"/>
      <j2 rule="demod" parents="31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="182">
    <literal><![CDATA[
      x + (x * 0) = x
    ]]></literal>
    <justification jstring="[para(10(a,1),32(a,1,2)),demod(17(2)),flip(a)].">
      <j1 rule="para" parents="10 32"/>
      <j2 rule="demod" parents="17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      x + (x * - (x * x)) = x * 0
    ]]></literal>
    <justification jstring="[para(11(a,1),32(a,1,2)),flip(a)].">
      <j1 rule="para" parents="11 32"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="213">
    <literal><![CDATA[
      x * (y + (0 * y)) = x * y
    ]]></literal>
    <justification jstring="[para(182(a,1),15(a,2,1)),demod(16(4),14(5))].">
      <j1 rule="para" parents="182 15"/>
      <j2 rule="demod" parents="16 14"/>
    </justification>
  </clause>

  <clause id="214">
    <literal><![CDATA[
      0 * 0 = 0
    ]]></literal>
    <justification jstring="[para(182(a,1),24(a,1)),flip(a)].">
      <j1 rule="para" parents="182 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="220">
    <literal><![CDATA[
      x + (y * 0) = x
    ]]></literal>
    <justification jstring="[para(182(a,1),42(a,1,2,2)),demod(23(3)),flip(a)].">
      <j1 rule="para" parents="182 42"/>
      <j2 rule="demod" parents="23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      x * 0 = 0
    ]]></literal>
    <justification jstring="[para(214(a,1),15(a,1,1)),demod(220(4),24(3)),flip(a)].">
      <j1 rule="para" parents="214 15"/>
      <j2 rule="demod" parents="220 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="224">
    <literal><![CDATA[
      x + (x * - (x * x)) = 0
    ]]></literal>
    <justification jstring="[back_demod(183),demod(222(6))].">
      <j1 rule="back_demod" parents="183"/>
      <j2 rule="demod" parents="222"/>
    </justification>
  </clause>

  <clause id="228">
    <literal><![CDATA[
      (x + (y * z)) * (z * z) = ((x * z) + y) * z
    ]]></literal>
    <justification jstring="[para(17(a,1),31(a,1,2,2)),demod(30(4)),flip(a)].">
      <j1 rule="para" parents="17 31"/>
      <j2 rule="demod" parents="30"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="257">
    <literal><![CDATA[
      x * (y + (- (x * x) * y)) = 0 * y
    ]]></literal>
    <justification jstring="[para(224(a,1),15(a,2,1)),demod(16(5),14(6))].">
      <j1 rule="para" parents="224 15"/>
      <j2 rule="demod" parents="16 14"/>
    </justification>
  </clause>

  <clause id="259">
    <literal><![CDATA[
      - (x * - (x * x)) = x
    ]]></literal>
    <justification jstring="[para(224(a,1),23(a,1,2)),demod(13(6),24(6))].">
      <j1 rule="para" parents="224 23"/>
      <j2 rule="demod" parents="13 24"/>
    </justification>
  </clause>

  <clause id="272">
    <literal><![CDATA[
      x * - (x * x) = - x
    ]]></literal>
    <justification jstring="[para(259(a,1),41(a,1,1)),flip(a)].">
      <j1 rule="para" parents="259 41"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="285">
    <literal><![CDATA[
      x * (x * - x) = - x
    ]]></literal>
    <justification jstring="[para(272(a,1),35(a,1,2,2)),demod(272(6))].">
      <j1 rule="para" parents="272 35"/>
      <j2 rule="demod" parents="272"/>
    </justification>
  </clause>

  <clause id="311">
    <literal><![CDATA[
      x * (x * (- x * y)) = - x * y
    ]]></literal>
    <justification jstring="[para(285(a,1),16(a,1,1)),demod(16(5)),flip(a)].">
      <j1 rule="para" parents="285 16"/>
      <j2 rule="demod" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="396">
    <literal><![CDATA[
      x * ((y + (0 * y)) * z) = x * (y * z)
    ]]></literal>
    <justification jstring="[para(213(a,1),16(a,1,1)),demod(16(2)),flip(a)].">
      <j1 rule="para" parents="213 16"/>
      <j2 rule="demod" parents="16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="397">
    <literal><![CDATA[
      x + (0 * x) = x
    ]]></literal>
    <justification jstring="[para(213(a,1),17(a,1,2)),demod(396(8),228(5),13(3),24(3),16(2),17(2)),flip(a)].">
      <j1 rule="para" parents="213 17"/>
      <j2 rule="demod" parents="396 228 13 24 16 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="401">
    <literal><![CDATA[
      0 * - x = 0
    ]]></literal>
    <justification jstring="[para(397(a,1),26(a,1,2)),demod(11(2)),flip(a)].">
      <j1 rule="para" parents="397 26"/>
      <j2 rule="demod" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="416">
    <literal><![CDATA[
      0 * x = 0
    ]]></literal>
    <justification jstring="[para(41(a,1),401(a,1,2))].">
      <j1 rule="para" parents="41 401"/>
    </justification>
  </clause>

  <clause id="418">
    <literal><![CDATA[
      x * (y + (- (x * x) * y)) = 0
    ]]></literal>
    <justification jstring="[back_demod(257),demod(416(7))].">
      <j1 rule="back_demod" parents="257"/>
      <j2 rule="demod" parents="416"/>
    </justification>
  </clause>

  <clause id="437">
    <literal><![CDATA[
      x * ((y + (- (x * x) * y)) * z) = 0
    ]]></literal>
    <justification jstring="[para(418(a,1),16(a,1,1)),demod(416(2)),flip(a)].">
      <j1 rule="para" parents="418 16"/>
      <j2 rule="demod" parents="416"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="498">
    <literal><![CDATA[
      - (x * y) = x * - y
    ]]></literal>
    <justification jstring="[para(23(a,1),44(a,1,2,2)),demod(13(5),66(5))].">
      <j1 rule="para" parents="23 44"/>
      <j2 rule="demod" parents="13 66"/>
    </justification>
  </clause>

  <clause id="507">
    <literal><![CDATA[
      - x * - y = x * y
    ]]></literal>
    <justification jstring="[para(418(a,1),44(a,1,2)),demod(498(2),16(3),311(4),498(3),13(5),24(5))].">
      <j1 rule="para" parents="418 44"/>
      <j2 rule="demod" parents="498 16 311 498 13 24"/>
    </justification>
  </clause>

  <clause id="535">
    <literal><![CDATA[
      x * ((y + (x * (- x * y))) * z) = 0
    ]]></literal>
    <justification jstring="[back_demod(437),demod(498(2),16(3))].">
      <j1 rule="back_demod" parents="437"/>
      <j2 rule="demod" parents="498 16"/>
    </justification>
  </clause>

  <clause id="575">
    <literal><![CDATA[
      (x * - y) + (z * y) = (x + - z) * - y
    ]]></literal>
    <justification jstring="[para(507(a,1),15(a,1,2))].">
      <j1 rule="para" parents="507 15"/>
    </justification>
  </clause>

  <clause id="578">
    <literal><![CDATA[
      - x * y = x * - y
    ]]></literal>
    <justification jstring="[para(41(a,1),507(a,1,1)),flip(a)].">
      <j1 rule="para" parents="41 507"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="585">
    <literal><![CDATA[
      (x * y) + (z * (u * - y)) = (- x + (z * u)) * - y
    ]]></literal>
    <justification jstring="[para(507(a,1),31(a,1,1))].">
      <j1 rule="para" parents="507 31"/>
    </justification>
  </clause>

  <clause id="596">
    <literal><![CDATA[
      x * ((y + (x * (x * - y))) * z) = 0
    ]]></literal>
    <justification jstring="[back_demod(535),demod(578(2))].">
      <j1 rule="back_demod" parents="535"/>
      <j2 rule="demod" parents="578"/>
    </justification>
  </clause>

  <clause id="614">
    <literal><![CDATA[
      (- x + y) * z = (x + - y) * - z
    ]]></literal>
    <justification jstring="[para(578(a,1),15(a,1,1)),demod(575(4)),flip(a)].">
      <j1 rule="para" parents="578 15"/>
      <j2 rule="demod" parents="575"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="618">
    <literal><![CDATA[
      (x * y) + (z * (u * - y)) = (x + (z * - u)) * y
    ]]></literal>
    <justification jstring="[back_demod(585),demod(614(10),498(7),41(10))].">
      <j1 rule="back_demod" parents="585"/>
      <j2 rule="demod" parents="614 498 41"/>
    </justification>
  </clause>

  <clause id="658">
    <literal><![CDATA[
      x + ((y + (x * z)) * x) = (y + (x * (x + z))) * x
    ]]></literal>
    <justification jstring="[para(50(a,1),163(a,1,1)),flip(a)].">
      <j1 rule="para" parents="50 163"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      (x + (y * (y * - x))) * y = 0
    ]]></literal>
    <justification jstring="[para(596(a,1),36(a,1,2,2,2)),demod(222(10),222(6),222(6)),flip(a)].">
      <j1 rule="para" parents="596 36"/>
      <j2 rule="demod" parents="222 222 222"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="698">
    <literal><![CDATA[
      (x * y) + (y * (y * (x * - y))) = 0
    ]]></literal>
    <justification jstring="[para(689(a,1),15(a,2)),demod(16(5),16(4),578(3))].">
      <j1 rule="para" parents="689 15"/>
      <j2 rule="demod" parents="16 16 578"/>
    </justification>
  </clause>

  <clause id="1411">
    <literal><![CDATA[
      x * (x * (y * x)) = y * x
    ]]></literal>
    <justification jstring="[para(698(a,1),23(a,1,2)),demod(498(5),498(4),498(3),41(2),13(5),24(5))].">
      <j1 rule="para" parents="698 23"/>
      <j2 rule="demod" parents="498 498 498 41 13 24"/>
    </justification>
  </clause>

  <clause id="1443">
    <literal><![CDATA[
      x * (x * (y * (x * z))) = y * (x * z)
    ]]></literal>
    <justification jstring="[para(1411(a,1),16(a,1,1)),demod(16(2),16(5),16(4)),flip(a)].">
      <j1 rule="para" parents="1411 16"/>
      <j2 rule="demod" parents="16 16 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1459">
    <literal><![CDATA[
      x * (y * (x * (y * (y * x)))) = y * x
    ]]></literal>
    <justification jstring="[para(36(a,1),1411(a,1,2,2)),demod(16(10),16(9),16(8),16(7),36(9),16(7),16(6),16(5),16(4),1411(3),36(10))].">
      <j1 rule="para" parents="36 1411"/>
      <j2 rule="demod" parents="16 16 16 16 36 16 16 16 16 1411 36"/>
    </justification>
  </clause>

  <clause id="1461">
    <literal><![CDATA[
      x * (x * (y * - x)) = y * - x
    ]]></literal>
    <justification jstring="[para(1411(a,1),498(a,1,1)),demod(498(2),498(5),498(4)),flip(a)].">
      <j1 rule="para" parents="1411 498"/>
      <j2 rule="demod" parents="498 498 498"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1462">
    <literal><![CDATA[
      x * (y * (z + (x * (x * - z)))) = 0
    ]]></literal>
    <justification jstring="[para(1411(a,1),596(a,1,2))].">
      <j1 rule="para" parents="1411 596"/>
    </justification>
  </clause>

  <clause id="1518">
    <literal><![CDATA[
      x * (y + (x * (z * - x))) = (x * y) + (z * - x)
    ]]></literal>
    <justification jstring="[para(1461(a,1),14(a,1,2)),flip(a)].">
      <j1 rule="para" parents="1461 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1777">
    <literal><![CDATA[
      x * (y + (y * (x * (x * (y * - y))))) = 0
    ]]></literal>
    <justification jstring="[para(32(a,1),1462(a,1,2)),demod(498(2))].">
      <j1 rule="para" parents="32 1462"/>
      <j2 rule="demod" parents="498"/>
    </justification>
  </clause>

  <clause id="4634">
    <literal><![CDATA[
      x * (y * (x * (y * x))) = y * (y * x)
    ]]></literal>
    <justification jstring="[para(36(a,1),1443(a,1,2)),flip(a)].">
      <j1 rule="para" parents="36 1443"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5176">
    <literal><![CDATA[
      x * (y * (x * (x * y))) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(1411(a,1),1459(a,1,2,2,2)),demod(16(4),4634(5),16(4))].">
      <j1 rule="para" parents="1411 1459"/>
      <j2 rule="demod" parents="16 4634 16"/>
    </justification>
  </clause>

  <clause id="7491">
    <literal><![CDATA[
      x * (y * (x * (x * (y * z)))) = y * (x * (y * z))
    ]]></literal>
    <justification jstring="[para(5176(a,1),16(a,1,1)),demod(16(3),16(2),16(7),16(6),16(5)),flip(a)].">
      <j1 rule="para" parents="5176 16"/>
      <j2 rule="demod" parents="16 16 16 16 16"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9675">
    <literal><![CDATA[
      (x * y) + (y * (x * (y * - y))) = 0
    ]]></literal>
    <justification jstring="[para(1777(a,1),14(a,2)),demod(7491(7))].">
      <j1 rule="para" parents="1777 14"/>
      <j2 rule="demod" parents="7491"/>
    </justification>
  </clause>

  <clause id="9730">
    <literal><![CDATA[
      (x + (y * (x * - y))) * y = 0
    ]]></literal>
    <justification jstring="[para(163(a,1),9675(a,1,1)),demod(179(7),12(8),618(7),68(3),498(2),13(4),658(8),26(5))].">
      <j1 rule="para" parents="163 9675"/>
      <j2 rule="demod" parents="179 12 618 68 498 13 658 26"/>
    </justification>
  </clause>

  <clause id="10335">
    <literal><![CDATA[
      (x + (y * (x * - y))) * (y * z) = 0
    ]]></literal>
    <justification jstring="[para(9730(a,1),16(a,1,1)),demod(416(2)),flip(a)].">
      <j1 rule="para" parents="9730 16"/>
      <j2 rule="demod" parents="416"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11135">
    <literal><![CDATA[
      (x * y) + (y * - x) = 0
    ]]></literal>
    <justification jstring="[para(10335(a,1),1411(a,1,2)),demod(222(6),1518(6)),flip(a)].">
      <j1 rule="para" parents="10335 1411"/>
      <j2 rule="demod" parents="222 1518"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="11191">
    <literal><![CDATA[
      x * y = y * x
    ]]></literal>
    <justification jstring="[para(11135(a,1),23(a,1,2)),demod(498(3),41(2),13(3),24(3))].">
      <j1 rule="para" parents="11135 23"/>
      <j2 rule="demod" parents="498 41 13 24"/>
    </justification>
  </clause>

  <clause id="11192">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(11191,a,18,a)].">
      <j1 rule="resolve" parents="11191 18"/>
    </justification>
  </clause>

</proof>

</proofs>
