<?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>identity.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3319 was started by mccune on cleo.thornwood,
Wed Nov 22 11:22:01 2006
The command was "/home/mccune/bin/prover9 -f identity.in".
]]></heading>

<proof number="1" length="88" max_count="51">

<comments><![CDATA[
% Proof 1 at 3.85 (+ 0.00) seconds.
% Length of proof is 88.
% Level of proof is 18.
% Maximum clause weight is 51.
% Given clauses 92.
]]></comments>

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

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

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

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

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

  <clause id="6" type="deny">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

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

  <clause id="8">
    <literal><![CDATA[
      ((x * y) * x) * (e * y) = x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),2(a,1))].">
      <j1 rule="para" parents="4 2"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      ((x * e) * y) * (x * z) = x * (y * z)
    ]]></literal>
    <justification jstring="[para(2(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="2 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

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

  <clause id="11">
    <literal><![CDATA[
      (x * y) * ((x * z) * (y * u)) = z * u
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,2))].">
      <j1 rule="para" parents="4 3"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (x * y) * ((x * z) * u) = z * (y * u)
    ]]></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="13">
    <literal><![CDATA[
      (x * y) * (z * (y * u)) = (x * z) * u
    ]]></literal>
    <justification jstring="[para(3(a,1),4(a,1,2)),flip(a)].">
      <j1 rule="para" parents="3 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      ((x * x) * x) * e = x
    ]]></literal>
    <justification jstring="[para(5(a,1),3(a,1,2))].">
      <j1 rule="para" parents="5 3"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      (((x * y) * x) * ((x * y) * y)) * e = x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),16(a,1,1))].">
      <j1 rule="para" parents="4 16"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      (x * y) * (e * (y * x)) = x
    ]]></literal>
    <justification jstring="[para(3(a,1),8(a,1,1,1)),rewrite(3(7))].">
      <j1 rule="para" parents="3 8"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      (x * y) * x = e * (y * x)
    ]]></literal>
    <justification jstring="[para(30(a,1),3(a,1,2))].">
      <j1 rule="para" parents="30 3"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x * (e * (x * x)) = e * (x * x)
    ]]></literal>
    <justification jstring="[para(5(a,1),30(a,1,1)),rewrite(36(4),3(8),36(6))].">
      <j1 rule="para" parents="5 30"/>
      <j2 rule="rewrite" parents="36 3 36"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      e * ((x * x) * e) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),30(a,1,2,2)),rewrite(36(2),41(4),7(6),36(5))].">
      <j1 rule="para" parents="5 30"/>
      <j2 rule="rewrite" parents="36 41 7 36"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      (x * x) * (e * x) = e
    ]]></literal>
    <justification jstring="[para(16(a,1),30(a,1,2,2)),rewrite(36(3),3(5))].">
      <j1 rule="para" parents="16 30"/>
      <j2 rule="rewrite" parents="36 3"/>
    </justification>
  </clause>

  <clause id="51">
    <literal><![CDATA[
      ((e * (x * y)) * ((y * x) * x)) * e = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite(36(2))].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (e * (x * x)) * x = e
    ]]></literal>
    <justification jstring="[back_rewrite(5),rewrite(36(2))].">
      <j1 rule="back_rewrite" parents="5"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      (x * x) * e = e * x
    ]]></literal>
    <justification jstring="[para(42(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="42 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      ((x * e) * y) * z = x * (y * (x * z))
    ]]></literal>
    <justification jstring="[para(3(a,1),9(a,1,2))].">
      <j1 rule="para" parents="3 9"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      (e * ((x * (y * e)) * y)) * (y * (x * (y * z))) = y * (x * z)
    ]]></literal>
    <justification jstring="[para(9(a,1),9(a,2)),rewrite(61(5),36(5),61(10))].">
      <j1 rule="para" parents="9 9"/>
      <j2 rule="rewrite" parents="61 36 61"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      (x * e) * (x * x) = e
    ]]></literal>
    <justification jstring="[para(43(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="43 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      (e * x) * (x * x) = x * x
    ]]></literal>
    <justification jstring="[para(43(a,1),30(a,1,1)),rewrite(3(8))].">
      <j1 rule="para" parents="43 30"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      ((x * x) * y) * (e * z) = (e * x) * (y * z)
    ]]></literal>
    <justification jstring="[para(58(a,1),4(a,1,1)),flip(a)].">
      <j1 rule="para" parents="58 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      (x * e) * e = x * x
    ]]></literal>
    <justification jstring="[para(73(a,1),3(a,1,2))].">
      <j1 rule="para" parents="73 3"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      (x * (e * (x * e))) * (x * (e * x)) = e
    ]]></literal>
    <justification jstring="[para(10(a,2),43(a,1)),rewrite(36(4))].">
      <j1 rule="para" parents="10 43"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="138">
    <literal><![CDATA[
      (x * x) * y = x * ((y * e) * (x * (e * y)))
    ]]></literal>
    <justification jstring="[para(82(a,1),10(a,2,1)),rewrite(61(8)),flip(a)].">
      <j1 rule="para" parents="82 10"/>
      <j2 rule="rewrite" parents="61"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      (x * x) * (x * (y * (x * (e * z)))) = y * z
    ]]></literal>
    <justification jstring="[para(82(a,1),11(a,1,1)),rewrite(61(7))].">
      <j1 rule="para" parents="82 11"/>
      <j2 rule="rewrite" parents="61"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      (e * (x * y)) * ((x * x) * (y * y)) = (x * y) * (x * y)
    ]]></literal>
    <justification jstring="[para(4(a,1),75(a,1,2))].">
      <j1 rule="para" parents="4 75"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      e * (x * y) = z * (x * (e * (z * y)))
    ]]></literal>
    <justification jstring="[para(30(a,1),12(a,1,2)),rewrite(36(2))].">
      <j1 rule="para" parents="30 12"/>
      <j2 rule="rewrite" parents="36"/>
    </justification>
  </clause>

  <clause id="212">
    <literal><![CDATA[
      (x * y) * e = x * (y * (e * x))
    ]]></literal>
    <justification jstring="[para(43(a,1),12(a,1,2))].">
      <j1 rule="para" parents="43 12"/>
    </justification>
  </clause>

  <clause id="239">
    <literal><![CDATA[
      ((x * y) * z) * (((x * (u * e)) * (y * u)) * v) = u * (z * v)
    ]]></literal>
    <justification jstring="[para(10(a,2),12(a,1,2,1))].">
      <j1 rule="para" parents="10 12"/>
    </justification>
  </clause>

  <clause id="255">
    <literal><![CDATA[
      (e * (x * y)) * (((y * x) * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(36(a,1),12(a,1,1))].">
      <j1 rule="para" parents="36 12"/>
    </justification>
  </clause>

  <clause id="271">
    <literal><![CDATA[
      ((x * y) * ((x * e) * z)) * ((y * z) * (y * z)) = (y * z) * (y * z)
    ]]></literal>
    <justification jstring="[para(12(a,2),75(a,1,1))].">
      <j1 rule="para" parents="12 75"/>
    </justification>
  </clause>

  <clause id="312">
    <literal><![CDATA[
      x * (y * (x * y)) = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(51),rewrite(212(8),3(10),255(8))].">
      <j1 rule="back_rewrite" parents="51"/>
      <j2 rule="rewrite" parents="212 3 255"/>
    </justification>
  </clause>

  <clause id="314">
    <literal><![CDATA[
      (e * x) * (x * (e * x)) = e
    ]]></literal>
    <justification jstring="[back_rewrite(108),rewrite(312(5))].">
      <j1 rule="back_rewrite" parents="108"/>
      <j2 rule="rewrite" parents="312"/>
    </justification>
  </clause>

  <clause id="327">
    <literal><![CDATA[
      x * (y * x) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(312(a,1),3(a,1,2))].">
      <j1 rule="para" parents="312 3"/>
    </justification>
  </clause>

  <clause id="368">
    <literal><![CDATA[
      x * ((e * x) * x) = e
    ]]></literal>
    <justification jstring="[back_rewrite(314),rewrite(327(6))].">
      <j1 rule="back_rewrite" parents="314"/>
      <j2 rule="rewrite" parents="327"/>
    </justification>
  </clause>

  <clause id="372">
    <literal><![CDATA[
      (e * x) * x = x * e
    ]]></literal>
    <justification jstring="[para(368(a,1),3(a,1,2)),flip(a)].">
      <j1 rule="para" parents="368 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="380">
    <literal><![CDATA[
      x * (y * (e * x)) = z * (y * (x * (z * (e * x))))
    ]]></literal>
    <justification jstring="[para(368(a,1),12(a,1,2)),rewrite(212(3),372(9),212(7))].">
      <j1 rule="para" parents="368 12"/>
      <j2 rule="rewrite" parents="212 372 212"/>
    </justification>
  </clause>

  <clause id="397">
    <literal><![CDATA[
      (e * x) * y = e * (x * (e * y))
    ]]></literal>
    <justification jstring="[para(7(a,1),13(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="403">
    <literal><![CDATA[
      ((x * y) * z) * u = x * (z * (e * ((y * x) * (e * u))))
    ]]></literal>
    <justification jstring="[para(30(a,1),13(a,1,1)),rewrite(397(4)),flip(a)].">
      <j1 rule="para" parents="30 13"/>
      <j2 rule="rewrite" parents="397"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="406">
    <literal><![CDATA[
      e * (x * (y * z)) = y * (x * (e * z))
    ]]></literal>
    <justification jstring="[para(57(a,1),13(a,1,1)),rewrite(397(8),397(11),78(12),397(10),397(13),3(13),3(12))].">
      <j1 rule="para" parents="57 13"/>
      <j2 rule="rewrite" parents="397 397 78 397 397 3 3"/>
    </justification>
  </clause>

  <clause id="434">
    <literal><![CDATA[
      (x * y) * (z * (y * x)) = e * (z * x)
    ]]></literal>
    <justification jstring="[para(13(a,2),36(a,1))].">
      <j1 rule="para" parents="13 36"/>
    </justification>
  </clause>

  <clause id="442">
    <literal><![CDATA[
      x * (y * (e * ((z * x) * (e * (x * ((z * (u * v)) * (e * ((u * x) * (e * (y * w)))))))))) = v * w
    ]]></literal>
    <justification jstring="[para(13(a,2),11(a,1,2,1)),rewrite(403(8),403(14))].">
      <j1 rule="para" parents="13 11"/>
      <j2 rule="rewrite" parents="403 403"/>
    </justification>
  </clause>

  <clause id="444">
    <literal><![CDATA[
      (x * y) * (z * (y * ((x * u) * (z * v)))) = u * v
    ]]></literal>
    <justification jstring="[para(13(a,2),11(a,1))].">
      <j1 rule="para" parents="13 11"/>
    </justification>
  </clause>

  <clause id="516">
    <literal><![CDATA[
      e * ((x * y) * (e * (y * (z * (e * ((x * y) * (e * u))))))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_rewrite(255),rewrite(403(6),397(12))].">
      <j1 rule="back_rewrite" parents="255"/>
      <j2 rule="rewrite" parents="403 397"/>
    </justification>
  </clause>

  <clause id="521">
    <literal><![CDATA[
      e * ((x * y) * (e * ((x * x) * (y * y)))) = (x * y) * (x * y)
    ]]></literal>
    <justification jstring="[back_rewrite(185),rewrite(397(7))].">
      <j1 rule="back_rewrite" parents="185"/>
      <j2 rule="rewrite" parents="397"/>
    </justification>
  </clause>

  <clause id="532">
    <literal><![CDATA[
      e * (x * (y * (e * z))) = y * (x * z)
    ]]></literal>
    <justification jstring="[back_rewrite(70),rewrite(397(10),403(11),3(12),403(9),397(11),3(11),3(8),3(8),3(5),3(4))].">
      <j1 rule="back_rewrite" parents="70"/>
      <j2 rule="rewrite" parents="397 403 3 403 397 3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="564">
    <literal><![CDATA[
      x * (y * (e * ((z * y) * (e * ((z * x) * (z * x)))))) = (z * x) * (z * x)
    ]]></literal>
    <justification jstring="[back_rewrite(271),rewrite(403(9),403(13),3(14),397(11),3(14),3(13))].">
      <j1 rule="back_rewrite" parents="271"/>
      <j2 rule="rewrite" parents="403 403 3 397 3 3"/>
    </justification>
  </clause>

  <clause id="577">
    <literal><![CDATA[
      x * (y * (e * (x * (z * (e * u))))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_rewrite(239),rewrite(403(8),403(10),3(11),397(8),3(11),403(13),444(15),3(7))].">
      <j1 rule="back_rewrite" parents="239"/>
      <j2 rule="rewrite" parents="403 403 3 397 3 403 444 3"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      c4 * (c3 * (e * ((c2 * c4) * (e * c1)))) != c1 * (c3 * (e * ((c2 * c1) * (e * c4))))
    ]]></literal>
    <justification jstring="[back_rewrite(6),rewrite(403(7),403(20))].">
      <j1 rule="back_rewrite" parents="6"/>
      <j2 rule="rewrite" parents="403 403"/>
    </justification>
  </clause>

  <clause id="626">
    <literal><![CDATA[
      (x * x) * y = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_rewrite(138),rewrite(434(8))].">
      <j1 rule="back_rewrite" parents="138"/>
      <j2 rule="rewrite" parents="434"/>
    </justification>
  </clause>

  <clause id="627">
    <literal><![CDATA[
      x * (y * (e * ((z * u) * ((z * x) * (e * (y * v)))))) = u * v
    ]]></literal>
    <justification jstring="[back_rewrite(442),rewrite(444(16))].">
      <j1 rule="back_rewrite" parents="442"/>
      <j2 rule="rewrite" parents="444"/>
    </justification>
  </clause>

  <clause id="640">
    <literal><![CDATA[
      e * ((x * y) * (e * (x * (e * (x * (y * y)))))) = (x * y) * (x * y)
    ]]></literal>
    <justification jstring="[back_rewrite(521),rewrite(626(6))].">
      <j1 rule="back_rewrite" parents="521"/>
      <j2 rule="rewrite" parents="626"/>
    </justification>
  </clause>

  <clause id="653">
    <literal><![CDATA[
      x * (e * (y * (x * (e * z)))) = y * z
    ]]></literal>
    <justification jstring="[back_rewrite(171),rewrite(626(7),3(7))].">
      <j1 rule="back_rewrite" parents="171"/>
      <j2 rule="rewrite" parents="626 3"/>
    </justification>
  </clause>

  <clause id="663">
    <literal><![CDATA[
      c1 * (c3 * (e * ((c2 * c1) * (e * c4)))) != c4 * (c3 * (e * ((c2 * e) * (c4 * c1))))
    ]]></literal>
    <justification jstring="[para(4(a,1),624(a,1,2,2,2)),flip(a)].">
      <j1 rule="para" parents="4 624"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="677">
    <literal><![CDATA[
      c4 * (c3 * (e * ((c2 * e) * (c4 * c1)))) != c1 * (c3 * (e * ((c2 * e) * (c1 * c4))))
    ]]></literal>
    <justification jstring="[para(4(a,1),663(a,1,2,2,2)),flip(a)].">
      <j1 rule="para" parents="4 663"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="712">
    <literal><![CDATA[
      e * (x * y) = x * (y * (e * (y * x)))
    ]]></literal>
    <justification jstring="[para(327(a,1),13(a,1)),rewrite(626(2),36(7)),flip(a)].">
      <j1 rule="para" parents="327 13"/>
      <j2 rule="rewrite" parents="626 36"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="738">
    <literal><![CDATA[
      (x * y) * z = e * (x * (e * (x * (e * (y * (e * (x * (e * (x * (e * z))))))))))
    ]]></literal>
    <justification jstring="[para(41(a,1),13(a,1,1)),rewrite(397(7),626(8),397(13),626(14)),flip(a)].">
      <j1 rule="para" parents="41 13"/>
      <j2 rule="rewrite" parents="397 626 397 626"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="744">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * (e * (x * (e * (x * (e * (x * y))))))))))) = x * (e * (x * (y * y)))
    ]]></literal>
    <justification jstring="[back_rewrite(640),rewrite(738(10),3(15),3(12),3(11),3(8),41(8),3(8),3(9),738(8)),flip(a)].">
      <j1 rule="back_rewrite" parents="640"/>
      <j2 rule="rewrite" parents="738 3 3 3 3 41 3 3 738"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="745">
    <literal><![CDATA[
      x * (y * (z * (e * (z * (e * (y * (e * (x * x)))))))) = z * (e * (z * (x * x)))
    ]]></literal>
    <justification jstring="[back_rewrite(564),rewrite(738(6),744(21),738(10),3(15),3(12),3(11),3(8),3(14),738(15),744(30))].">
      <j1 rule="back_rewrite" parents="564"/>
      <j2 rule="rewrite" parents="738 744 738 3 3 3 3 3 738 744"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      c4 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c4 * c1)))))))))) != c1 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c1 * c4))))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(677),rewrite(738(10),3(23),3(25),738(33),3(46),3(48))].">
      <j1 rule="back_rewrite" parents="677"/>
      <j2 rule="rewrite" parents="738 3 3 738 3 3"/>
    </justification>
  </clause>

  <clause id="815">
    <literal><![CDATA[
      x * (y * (z * (e * (z * (e * (u * (x * (e * (z * (e * (z * (y * v)))))))))))) = u * v
    ]]></literal>
    <justification jstring="[back_rewrite(627),rewrite(738(7),3(12),738(19),3(24),3(21),3(20),3(17),3(16),3(19))].">
      <j1 rule="back_rewrite" parents="627"/>
      <j2 rule="rewrite" parents="738 3 738 3 3 3 3 3 3"/>
    </justification>
  </clause>

  <clause id="896">
    <literal><![CDATA[
      x * (e * (x * (e * (y * (e * (x * (e * (x * (y * (z * (x * (e * (x * (e * (y * (e * (x * (e * (x * u))))))))))))))))))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_rewrite(516),rewrite(738(8),3(13),3(20),738(20),3(25),3(32))].">
      <j1 rule="back_rewrite" parents="516"/>
      <j2 rule="rewrite" parents="738 3 3 738 3 3"/>
    </justification>
  </clause>

  <clause id="1140">
    <literal><![CDATA[
      e * (x * (y * e)) = y * (x * e)
    ]]></literal>
    <justification jstring="[para(7(a,1),406(a,2,2,2))].">
      <j1 rule="para" parents="7 406"/>
    </justification>
  </clause>

  <clause id="1151">
    <literal><![CDATA[
      e * (x * (y * x)) = x * (y * (e * y))
    ]]></literal>
    <justification jstring="[para(327(a,1),406(a,1,2))].">
      <j1 rule="para" parents="327 406"/>
    </justification>
  </clause>

  <clause id="1161">
    <literal><![CDATA[
      e * (x * (y * z)) = u * (x * (y * (u * (e * z))))
    ]]></literal>
    <justification jstring="[para(406(a,1),207(a,2,2,2))].">
      <j1 rule="para" parents="406 207"/>
    </justification>
  </clause>

  <clause id="1191">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * (y * (e * (x * e))))))))))) = x * (y * (x * e))
    ]]></literal>
    <justification jstring="[para(312(a,1),380(a,2,2,2,2)),rewrite(738(8),3(18),738(24),3(29),41(28),3(28),3(26),3(23),312(24))].">
      <j1 rule="para" parents="312 380"/>
      <j2 rule="rewrite" parents="738 3 738 3 41 3 3 3 312"/>
    </justification>
  </clause>

  <clause id="1216">
    <literal><![CDATA[
      x * (e * (y * (x * (y * e)))) = e * y
    ]]></literal>
    <justification jstring="[para(1140(a,1),380(a,1,2)),rewrite(7(5),738(5),312(11),3(9),3(8),41(6),3(6),3(3),738(11),3(21),1191(22)),flip(a)].">
      <j1 rule="para" parents="1140 380"/>
      <j2 rule="rewrite" parents="7 738 312 3 3 41 3 3 738 3 1191"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1310">
    <literal><![CDATA[
      e * (x * (y * (e * (y * x)))) = x * y
    ]]></literal>
    <justification jstring="[para(532(a,2),312(a,1))].">
      <j1 rule="para" parents="532 312"/>
    </justification>
  </clause>

  <clause id="1321">
    <literal><![CDATA[
      e * (x * (y * (e * (z * (e * (x * u)))))) = y * (e * (z * u))
    ]]></literal>
    <justification jstring="[para(207(a,2),532(a,2,2))].">
      <j1 rule="para" parents="207 532"/>
    </justification>
  </clause>

  <clause id="1325">
    <literal><![CDATA[
      c4 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c4 * c1)))))))))) != e * (c3 * (c1 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c1 * c4))))))))))))
    ]]></literal>
    <justification jstring="[para(532(a,2),791(a,2))].">
      <j1 rule="para" parents="532 791"/>
    </justification>
  </clause>

  <clause id="1905">
    <literal><![CDATA[
      e * (x * (y * (z * (x * (e * u))))) = y * (z * u)
    ]]></literal>
    <justification jstring="[para(653(a,1),406(a,2,2))].">
      <j1 rule="para" parents="653 406"/>
    </justification>
  </clause>

  <clause id="1924">
    <literal><![CDATA[
      x * (e * (x * (e * (y * (z * (e * (x * (e * (x * u))))))))) = y * (z * u)
    ]]></literal>
    <justification jstring="[back_rewrite(896),rewrite(1905(21),1905(17),3(12))].">
      <j1 rule="back_rewrite" parents="896"/>
      <j2 rule="rewrite" parents="1905 1905 3"/>
    </justification>
  </clause>

  <clause id="1932">
    <literal><![CDATA[
      x * (y * (z * (x * (y * u)))) = z * u
    ]]></literal>
    <justification jstring="[back_rewrite(815),rewrite(1924(15))].">
      <j1 rule="back_rewrite" parents="815"/>
      <j2 rule="rewrite" parents="1924"/>
    </justification>
  </clause>

  <clause id="1950">
    <literal><![CDATA[
      x * (e * (x * (e * (x * (e * (x * (e * y))))))) = e * (x * (e * (x * (e * (x * y)))))
    ]]></literal>
    <justification jstring="[para(1140(a,1),712(a,2,2,2)),rewrite(738(4),3(14),3(16),738(19),1216(25),3(21),3(20))].">
      <j1 rule="para" parents="1140 712"/>
      <j2 rule="rewrite" parents="738 3 3 738 1216 3 3"/>
    </justification>
  </clause>

  <clause id="2127">
    <literal><![CDATA[
      c4 * (c3 * (e * (c2 * (e * (c2 * (e * (c2 * (c4 * c1)))))))) != e * (c3 * (c1 * (c2 * (e * (c2 * (e * (c2 * (c1 * c4))))))))
    ]]></literal>
    <justification jstring="[back_rewrite(1325),rewrite(1950(21),1950(42),3(39))].">
      <j1 rule="back_rewrite" parents="1325"/>
      <j2 rule="rewrite" parents="1950 1950 3"/>
    </justification>
  </clause>

  <clause id="2331">
    <literal><![CDATA[
      c4 * (c3 * (e * (c2 * (e * (c2 * (c4 * (c2 * (e * c1)))))))) != e * (c3 * (c1 * (c2 * (e * (c2 * (e * (c2 * (c1 * c4))))))))
    ]]></literal>
    <justification jstring="[para(577(a,2),2127(a,1,2,2,2,2,2,2,2)),rewrite(1932(19))].">
      <j1 rule="para" parents="577 2127"/>
      <j2 rule="rewrite" parents="1932"/>
    </justification>
  </clause>

  <clause id="2709">
    <literal><![CDATA[
      c4 * (c3 * (e * (c2 * (c4 * (c2 * (e * (c2 * (e * c1)))))))) != e * (c3 * (c1 * (c2 * (e * (c2 * (e * (c2 * (c1 * c4))))))))
    ]]></literal>
    <justification jstring="[para(577(a,2),2331(a,1,2,2,2,2,2)),rewrite(1932(21))].">
      <j1 rule="para" parents="577 2331"/>
      <j2 rule="rewrite" parents="1932"/>
    </justification>
  </clause>

  <clause id="2882">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * y))))))) = x * (e * (x * (e * (x * (e * y)))))
    ]]></literal>
    <justification jstring="[para(1151(a,2),712(a,2,2,2,2)),rewrite(738(5),738(15),3(23),3(21),3(19),3(16),3(15),3(13),3(11),3(13),3(18),3(15),738(14),738(24),3(32),3(30),3(28),3(25),3(24),3(22),3(20)),flip(a)].">
      <j1 rule="para" parents="1151 712"/>
      <j2 rule="rewrite" parents="738 738 3 3 3 3 3 3 3 3 3 3 738 738 3 3 3 3 3 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3873">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (y * (e * (y * (e * x))))))))) = y * (e * (y * x))
    ]]></literal>
    <justification jstring="[para(1151(a,2),745(a,1,2,2,2,2,2,2,2,2)),rewrite(3(14),738(13),3(18),41(17),3(17),738(14),3(23),3(21),41(19),3(19),3(17),3(15),41(13),3(13),3(10),3(10),738(14),1321(19),738(18),3(26),3(24),3(22),3(19),3(18),3(16),3(14),738(23),312(28),3(26),7(25),738(26),3(34),3(32),3(30),3(27),7(26),7(25),7(24),7(23),7(22),7(21),312(22),3(20))].">
      <j1 rule="para" parents="1151 745"/>
      <j2 rule="rewrite" parents="3 738 3 41 3 738 3 3 41 3 3 3 41 3 3 3 738 1321 738 3 3 3 3 3 3 3 738 312 3 7 738 3 3 3 3 7 7 7 7 7 7 312 3"/>
    </justification>
  </clause>

  <clause id="3892">
    <literal><![CDATA[
      c4 * (c3 * (c1 * (c4 * (e * (c2 * (e * (c2 * (e * (c2 * (c1 * c4)))))))))) != e * (c3 * (c1 * (c2 * (e * (c2 * (e * (c2 * (c1 * c4))))))))
    ]]></literal>
    <justification jstring="[para(712(a,1),2709(a,1,2,2)),rewrite(738(28),738(41),3(49),738(50),3(58),3(56),3(54),738(51),3873(61),1310(49),738(42),3(51),3(49),3(49),3(47),3(45),1932(43),3(38),3(36),3(34),3(36),3(37),738(34),2882(46),738(43),738(56),3(64),3(62),3(60),738(57),1950(68),3(65),738(62),3(71),3(69),3(69),3(67),3(65),1932(63),1932(57),3(51),3(49),3(47),1932(45),3(40),1905(38),1905(32),3(25))].">
      <j1 rule="para" parents="712 2709"/>
      <j2 rule="rewrite" parents="738 738 3 738 3 3 3 738 3873 1310 738 3 3 3 3 3 1932 3 3 3 3 3 738 2882 738 738 3 3 3 738 1950 3 738 3 3 3 3 3 1932 1932 3 3 3 1932 3 1905 1905 3"/>
    </justification>
  </clause>

  <clause id="3893">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(3892,a,1161,a(flip))].">
      <j1 rule="resolve" parents="3892 1161"/>
    </justification>
  </clause>

</proof>

</proofs>
