<?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="4">

<source>a1.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13092 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:11 2006
The command was "/home/mccune/bin/prover9 -f a1.in".
]]></heading>

<proof number="1" length="51" max_count="83">

<comments><![CDATA[
% Proof 1 at 2.25 (+ 0.03) seconds: McKenzie_2.
% Length of proof is 51.
% Level of proof is 27.
% Maximum clause weight is 83.
% Given clauses 140.
]]></comments>

  <clause id="6">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v ((v6 v y) ^ (y v v7))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      c4 ^ (c5 v (c4 v c6)) != c4
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_2)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (((x v ((y v z) ^ (z v u))) ^ ((y v z) ^ (z v u))) v ((z v (v ^ ((y v z) ^ (z v u)))) ^ w)) ^ (v6 v ((v7 v ((y v z) ^ (z v u))) ^ (((y v z) ^ (z v u)) v v8))) = (y v z) ^ (z v u)
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1,2,1,1))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1,2))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (((x v (((y ^ (z v z)) v (u ^ z)) ^ v)) ^ (((y ^ (z v z)) v (u ^ z)) ^ v)) v (((w ^ ((((y ^ (z v z)) v (u ^ z)) ^ v) v (((y ^ (z v z)) v (u ^ z)) ^ v))) v (v6 ^ (((y ^ (z v z)) v (u ^ z)) ^ v))) ^ v7)) ^ (v8 v z) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="13 13"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (v v ((w v y) ^ (y v v6))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),6(a,1,1,2,1,1))].">
      <j1 rule="para" parents="17 6"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (v v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),18(a,1,2,2))].">
      <j1 rule="para" parents="13 18"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(17(a,1),24(a,1,1,2,1,2))].">
      <j1 rule="para" parents="17 24"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (v6 v ((v7 v y) ^ (y v v8))) = y
    ]]></literal>
    <justification jstring="[para(18(a,1),11(a,1,1,1,1,2)),demod(18(17),18(22),18(26),18(27),18(31))].">
      <j1 rule="para" parents="18 11"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),34(a,1,1,2))].">
      <j1 rule="para" parents="6 34"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="13 44"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (v v z) = y ^ z
    ]]></literal>
    <justification jstring="[para(44(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="44 44"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(17(a,1),57(a,1,1,2))].">
      <j1 rule="para" parents="17 57"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),13(a,1,1,1))].">
      <j1 rule="para" parents="69 13"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),57(a,1,1,1))].">
      <j1 rule="para" parents="69 57"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),69(a,1,1,1))].">
      <j1 rule="para" parents="69 69"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(14(a,1),92(a,1,1,2))].">
      <j1 rule="para" parents="14 92"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(67(a,1),103(a,1,1,2,1,1))].">
      <j1 rule="para" parents="67 103"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      ((x v x) v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(6(a,1),148(a,1,1,2))].">
      <j1 rule="para" parents="6 148"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(171(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="171 6"/>
    </justification>
  </clause>

  <clause id="605">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(67(a,1),185(a,1,1,2,1,1))].">
      <j1 rule="para" parents="67 185"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),605(a,1,1,2))].">
      <j1 rule="para" parents="6 605"/>
    </justification>
  </clause>

  <clause id="652">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v y) = y
    ]]></literal>
    <justification jstring="[para(624(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="624 13"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v y) = y
    ]]></literal>
    <justification jstring="[para(624(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="624 44"/>
    </justification>
  </clause>

  <clause id="670">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),24(a,1,1,1))].">
      <j1 rule="para" parents="652 24"/>
    </justification>
  </clause>

  <clause id="672">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),28(a,1,1,1))].">
      <j1 rule="para" parents="652 28"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),69(a,1,1,1))].">
      <j1 rule="para" parents="652 69"/>
    </justification>
  </clause>

  <clause id="683">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),652(a,1,1,1)),demod(682(8))].">
      <j1 rule="para" parents="69 652"/>
      <j2 rule="demod" parents="682"/>
    </justification>
  </clause>

  <clause id="746">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(672),demod(683(3),683(3))].">
      <j1 rule="back_demod" parents="672"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(670),demod(683(3))].">
      <j1 rule="back_demod" parents="670"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="792">
    <literal><![CDATA[
      (x v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(94),demod(683(3))].">
      <j1 rule="back_demod" parents="94"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[back_demod(80),demod(683(4),683(3))].">
      <j1 rule="back_demod" parents="80"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="890">
    <literal><![CDATA[
      ((x v x) v ((y ^ x) v (y ^ x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(792(a,1),148(a,1,1,2))].">
      <j1 rule="para" parents="792 148"/>
    </justification>
  </clause>

  <clause id="1016">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ (z v (x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(683(a,1),746(a,1,1,2,1)),demod(683(10))].">
      <j1 rule="para" parents="683 746"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="1454">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ (z v (((x v y) ^ y) v ((x v y) ^ y))) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(44(a,1),1016(a,1,1,2))].">
      <j1 rule="para" parents="44 1016"/>
    </justification>
  </clause>

  <clause id="1649">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),748(a,1,1,2,1,2))].">
      <j1 rule="para" parents="652 748"/>
    </justification>
  </clause>

  <clause id="1705">
    <literal><![CDATA[
      (x v x) ^ (y v ((x v x) ^ ((x v x) v z))) = x v x
    ]]></literal>
    <justification jstring="[para(683(a,1),797(a,1,2,2,1))].">
      <j1 rule="para" parents="683 797"/>
    </justification>
  </clause>

  <clause id="1708">
    <literal><![CDATA[
      ((x ^ y) v (x ^ y)) ^ (z v (y v y)) = (x ^ y) v (x ^ y)
    ]]></literal>
    <justification jstring="[para(890(a,1),797(a,1,2,2))].">
      <j1 rule="para" parents="890 797"/>
    </justification>
  </clause>

  <clause id="1745">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(44(a,1),1705(a,1,2,2)),demod(657(7)),flip(a)].">
      <j1 rule="para" parents="44 1705"/>
      <j2 rule="demod" parents="657"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1794">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_demod(1454),demod(1745(5),1745(6),792(3),1745(6))].">
      <j1 rule="back_demod" parents="1454"/>
      <j2 rule="demod" parents="1745 1745 792 1745"/>
    </justification>
  </clause>

  <clause id="1822">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_demod(1745),demod(1794(5))].">
      <j1 rule="back_demod" parents="1745"/>
      <j2 rule="demod" parents="1794"/>
    </justification>
  </clause>

  <clause id="1839">
    <literal><![CDATA[
      (x ^ y) ^ (z v y) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(1708),demod(1794(3),1794(2),1794(6))].">
      <j1 rule="back_demod" parents="1708"/>
      <j2 rule="demod" parents="1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="1841">
    <literal><![CDATA[
      x ^ (y v (x ^ (x v z))) = x
    ]]></literal>
    <justification jstring="[back_demod(1705),demod(1794(1),1794(1),1794(1),1794(5))].">
      <j1 rule="back_demod" parents="1705"/>
      <j2 rule="demod" parents="1794 1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="1883">
    <literal><![CDATA[
      (x v (x ^ y)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_demod(1649),demod(1794(1),1794(1),1794(5))].">
      <j1 rule="back_demod" parents="1649"/>
      <j2 rule="demod" parents="1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="2334">
    <literal><![CDATA[
      x ^ (x ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[para(1794(a,1),1841(a,1,2))].">
      <j1 rule="para" parents="1794 1841"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(1883(a,1),1841(a,1,2,2)),demod(1883(4)),flip(a)].">
      <j1 rule="para" parents="1883 1841"/>
      <j2 rule="demod" parents="1883"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2341">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(2340(a,1),1822(a,1,1))].">
      <j1 rule="para" parents="2340 1822"/>
    </justification>
  </clause>

  <clause id="2345">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_demod(2334),demod(2341(3))].">
      <j1 rule="back_demod" parents="2334"/>
      <j2 rule="demod" parents="2341"/>
    </justification>
  </clause>

  <clause id="2362">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(2345(a,1),1839(a,1,1)),demod(2345(5))].">
      <j1 rule="para" parents="2345 1839"/>
      <j2 rule="demod" parents="2345"/>
    </justification>
  </clause>

  <clause id="2363">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_2)
    ]]></attribute>
    <justification jstring="[resolve(2362,a,8,a)].">
      <j1 rule="resolve" parents="2362 8"/>
    </justification>
  </clause>

</proof>

<proof number="2" length="56" max_count="83">

<comments><![CDATA[
% Proof 2 at 2.31 (+ 0.03) seconds: McKenzie_1.
% Length of proof is 56.
% Level of proof is 29.
% Maximum clause weight is 83.
% Given clauses 162.
]]></comments>

  <clause id="6">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v ((v6 v y) ^ (y v v7))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      c1 v (c2 ^ (c1 ^ c3)) != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_1)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (((x v ((y v z) ^ (z v u))) ^ ((y v z) ^ (z v u))) v ((z v (v ^ ((y v z) ^ (z v u)))) ^ w)) ^ (v6 v ((v7 v ((y v z) ^ (z v u))) ^ (((y v z) ^ (z v u)) v v8))) = (y v z) ^ (z v u)
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1,2,1,1))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1,2))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (((x v (((y ^ (z v z)) v (u ^ z)) ^ v)) ^ (((y ^ (z v z)) v (u ^ z)) ^ v)) v (((w ^ ((((y ^ (z v z)) v (u ^ z)) ^ v) v (((y ^ (z v z)) v (u ^ z)) ^ v))) v (v6 ^ (((y ^ (z v z)) v (u ^ z)) ^ v))) ^ v7)) ^ (v8 v z) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="13 13"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (v v ((w v y) ^ (y v v6))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),6(a,1,1,2,1,1))].">
      <j1 rule="para" parents="17 6"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (v v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),18(a,1,2,2))].">
      <j1 rule="para" parents="13 18"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(17(a,1),24(a,1,1,2,1,2))].">
      <j1 rule="para" parents="17 24"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (v6 v ((v7 v y) ^ (y v v8))) = y
    ]]></literal>
    <justification jstring="[para(18(a,1),11(a,1,1,1,1,2)),demod(18(17),18(22),18(26),18(27),18(31))].">
      <j1 rule="para" parents="18 11"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),34(a,1,1,2))].">
      <j1 rule="para" parents="6 34"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="13 44"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (v v z) = y ^ z
    ]]></literal>
    <justification jstring="[para(44(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="44 44"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(17(a,1),57(a,1,1,2))].">
      <j1 rule="para" parents="17 57"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),13(a,1,1,1))].">
      <j1 rule="para" parents="69 13"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),57(a,1,1,1))].">
      <j1 rule="para" parents="69 57"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),69(a,1,1,1))].">
      <j1 rule="para" parents="69 69"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(14(a,1),92(a,1,1,2))].">
      <j1 rule="para" parents="14 92"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ y)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(67(a,1),57(a,1,1,2))].">
      <j1 rule="para" parents="67 57"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(67(a,1),103(a,1,1,2,1,1))].">
      <j1 rule="para" parents="67 103"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      ((x v x) v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(6(a,1),148(a,1,1,2))].">
      <j1 rule="para" parents="6 148"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(171(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="171 6"/>
    </justification>
  </clause>

  <clause id="605">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(67(a,1),185(a,1,1,2,1,1))].">
      <j1 rule="para" parents="67 185"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),605(a,1,1,2))].">
      <j1 rule="para" parents="6 605"/>
    </justification>
  </clause>

  <clause id="652">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v y) = y
    ]]></literal>
    <justification jstring="[para(624(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="624 13"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v y) = y
    ]]></literal>
    <justification jstring="[para(624(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="624 44"/>
    </justification>
  </clause>

  <clause id="670">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),24(a,1,1,1))].">
      <j1 rule="para" parents="652 24"/>
    </justification>
  </clause>

  <clause id="672">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),28(a,1,1,1))].">
      <j1 rule="para" parents="652 28"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),69(a,1,1,1))].">
      <j1 rule="para" parents="652 69"/>
    </justification>
  </clause>

  <clause id="683">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),652(a,1,1,1)),demod(682(8))].">
      <j1 rule="para" parents="69 652"/>
      <j2 rule="demod" parents="682"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),112(a,1,1,1))].">
      <j1 rule="para" parents="652 112"/>
    </justification>
  </clause>

  <clause id="746">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(672),demod(683(3),683(3))].">
      <j1 rule="back_demod" parents="672"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(670),demod(683(3))].">
      <j1 rule="back_demod" parents="670"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="792">
    <literal><![CDATA[
      (x v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(94),demod(683(3))].">
      <j1 rule="back_demod" parents="94"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[back_demod(80),demod(683(4),683(3))].">
      <j1 rule="back_demod" parents="80"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="890">
    <literal><![CDATA[
      ((x v x) v ((y ^ x) v (y ^ x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(792(a,1),148(a,1,1,2))].">
      <j1 rule="para" parents="792 148"/>
    </justification>
  </clause>

  <clause id="1016">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ (z v (x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(683(a,1),746(a,1,1,2,1)),demod(683(10))].">
      <j1 rule="para" parents="683 746"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="1454">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ (z v (((x v y) ^ y) v ((x v y) ^ y))) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(44(a,1),1016(a,1,1,2))].">
      <j1 rule="para" parents="44 1016"/>
    </justification>
  </clause>

  <clause id="1649">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),748(a,1,1,2,1,2))].">
      <j1 rule="para" parents="652 748"/>
    </justification>
  </clause>

  <clause id="1705">
    <literal><![CDATA[
      (x v x) ^ (y v ((x v x) ^ ((x v x) v z))) = x v x
    ]]></literal>
    <justification jstring="[para(683(a,1),797(a,1,2,2,1))].">
      <j1 rule="para" parents="683 797"/>
    </justification>
  </clause>

  <clause id="1708">
    <literal><![CDATA[
      ((x ^ y) v (x ^ y)) ^ (z v (y v y)) = (x ^ y) v (x ^ y)
    ]]></literal>
    <justification jstring="[para(890(a,1),797(a,1,2,2))].">
      <j1 rule="para" parents="890 797"/>
    </justification>
  </clause>

  <clause id="1745">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(44(a,1),1705(a,1,2,2)),demod(657(7)),flip(a)].">
      <j1 rule="para" parents="44 1705"/>
      <j2 rule="demod" parents="657"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1794">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_demod(1454),demod(1745(5),1745(6),792(3),1745(6))].">
      <j1 rule="back_demod" parents="1454"/>
      <j2 rule="demod" parents="1745 1745 792 1745"/>
    </justification>
  </clause>

  <clause id="1822">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_demod(1745),demod(1794(5))].">
      <j1 rule="back_demod" parents="1745"/>
      <j2 rule="demod" parents="1794"/>
    </justification>
  </clause>

  <clause id="1839">
    <literal><![CDATA[
      (x ^ y) ^ (z v y) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(1708),demod(1794(3),1794(2),1794(6))].">
      <j1 rule="back_demod" parents="1708"/>
      <j2 rule="demod" parents="1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="1841">
    <literal><![CDATA[
      x ^ (y v (x ^ (x v z))) = x
    ]]></literal>
    <justification jstring="[back_demod(1705),demod(1794(1),1794(1),1794(1),1794(5))].">
      <j1 rule="back_demod" parents="1705"/>
      <j2 rule="demod" parents="1794 1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="1883">
    <literal><![CDATA[
      (x v (x ^ y)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_demod(1649),demod(1794(1),1794(1),1794(5))].">
      <j1 rule="back_demod" parents="1649"/>
      <j2 rule="demod" parents="1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="2202">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_demod(689),demod(1794(5))].">
      <j1 rule="back_demod" parents="689"/>
      <j2 rule="demod" parents="1794"/>
    </justification>
  </clause>

  <clause id="2334">
    <literal><![CDATA[
      x ^ (x ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[para(1794(a,1),1841(a,1,2))].">
      <j1 rule="para" parents="1794 1841"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(1883(a,1),1841(a,1,2,2)),demod(1883(4)),flip(a)].">
      <j1 rule="para" parents="1883 1841"/>
      <j2 rule="demod" parents="1883"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2341">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(2340(a,1),1822(a,1,1))].">
      <j1 rule="para" parents="2340 1822"/>
    </justification>
  </clause>

  <clause id="2345">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_demod(2334),demod(2341(3))].">
      <j1 rule="back_demod" parents="2334"/>
      <j2 rule="demod" parents="2341"/>
    </justification>
  </clause>

  <clause id="2376">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(2202(a,1),2345(a,1)),flip(a)].">
      <j1 rule="para" parents="2202 2345"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2446">
    <literal><![CDATA[
      (x v y) v (z ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(1839(a,1),2376(a,1,2))].">
      <j1 rule="para" parents="1839 2376"/>
    </justification>
  </clause>

  <clause id="2555">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2340(a,1),2446(a,1,1)),demod(2340(5))].">
      <j1 rule="para" parents="2340 2446"/>
      <j2 rule="demod" parents="2340"/>
    </justification>
  </clause>

  <clause id="2556">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_1)
    ]]></attribute>
    <justification jstring="[resolve(2555,a,7,a)].">
      <j1 rule="resolve" parents="2555 7"/>
    </justification>
  </clause>

</proof>

<proof number="3" length="57" max_count="89">

<comments><![CDATA[
% Proof 3 at 2.40 (+ 0.03) seconds: McKenzie_4.
% Length of proof is 57.
% Level of proof is 29.
% Maximum clause weight is 89.
% Given clauses 207.
]]></comments>

  <clause id="6">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v ((v6 v y) ^ (y v v7))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      ((c10 v c11) ^ (c11 v c12)) ^ c11 != c11
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_4)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (((x v ((y v z) ^ (z v u))) ^ ((y v z) ^ (z v u))) v ((z v (v ^ ((y v z) ^ (z v u)))) ^ w)) ^ (v6 v ((v7 v ((y v z) ^ (z v u))) ^ (((y v z) ^ (z v u)) v v8))) = (y v z) ^ (z v u)
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1,2,1,1))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1,2))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (((x v (((y ^ (z v z)) v (u ^ z)) ^ v)) ^ (((y ^ (z v z)) v (u ^ z)) ^ v)) v (((w ^ ((((y ^ (z v z)) v (u ^ z)) ^ v) v (((y ^ (z v z)) v (u ^ z)) ^ v))) v (v6 ^ (((y ^ (z v z)) v (u ^ z)) ^ v))) ^ v7)) ^ (v8 v z) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="13 13"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (v v ((w v y) ^ (y v v6))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),6(a,1,1,2,1,1))].">
      <j1 rule="para" parents="17 6"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (v v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),18(a,1,2,2))].">
      <j1 rule="para" parents="13 18"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(17(a,1),24(a,1,1,2,1,2))].">
      <j1 rule="para" parents="17 24"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      (((x v ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) ^ ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) v u) ^ (w v ((v6 v ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) ^ (((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v)) v v7))) = (y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v)
    ]]></literal>
    <justification jstring="[para(6(a,1),11(a,1,1,2))].">
      <j1 rule="para" parents="6 11"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (v6 v ((v7 v y) ^ (y v v8))) = y
    ]]></literal>
    <justification jstring="[para(18(a,1),11(a,1,1,1,1,2)),demod(18(17),18(22),18(26),18(27),18(31))].">
      <j1 rule="para" parents="18 11"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),34(a,1,1,2))].">
      <j1 rule="para" parents="6 34"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="13 44"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (v v z) = y ^ z
    ]]></literal>
    <justification jstring="[para(44(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="44 44"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(17(a,1),57(a,1,1,2))].">
      <j1 rule="para" parents="17 57"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),13(a,1,1,1))].">
      <j1 rule="para" parents="69 13"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),57(a,1,1,1))].">
      <j1 rule="para" parents="69 57"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),69(a,1,1,1))].">
      <j1 rule="para" parents="69 69"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(14(a,1),92(a,1,1,2))].">
      <j1 rule="para" parents="14 92"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ y)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(67(a,1),57(a,1,1,2))].">
      <j1 rule="para" parents="67 57"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(67(a,1),103(a,1,1,2,1,1))].">
      <j1 rule="para" parents="67 103"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      ((x v x) v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(6(a,1),148(a,1,1,2))].">
      <j1 rule="para" parents="6 148"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(171(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="171 6"/>
    </justification>
  </clause>

  <clause id="605">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(67(a,1),185(a,1,1,2,1,1))].">
      <j1 rule="para" parents="67 185"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),605(a,1,1,2))].">
      <j1 rule="para" parents="6 605"/>
    </justification>
  </clause>

  <clause id="652">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v y) = y
    ]]></literal>
    <justification jstring="[para(624(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="624 13"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v y) = y
    ]]></literal>
    <justification jstring="[para(624(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="624 44"/>
    </justification>
  </clause>

  <clause id="670">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),24(a,1,1,1))].">
      <j1 rule="para" parents="652 24"/>
    </justification>
  </clause>

  <clause id="672">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),28(a,1,1,1))].">
      <j1 rule="para" parents="652 28"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),69(a,1,1,1))].">
      <j1 rule="para" parents="652 69"/>
    </justification>
  </clause>

  <clause id="683">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),652(a,1,1,1)),demod(682(8))].">
      <j1 rule="para" parents="69 652"/>
      <j2 rule="demod" parents="682"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),112(a,1,1,1))].">
      <j1 rule="para" parents="652 112"/>
    </justification>
  </clause>

  <clause id="746">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(672),demod(683(3),683(3))].">
      <j1 rule="back_demod" parents="672"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(670),demod(683(3))].">
      <j1 rule="back_demod" parents="670"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="792">
    <literal><![CDATA[
      (x v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(94),demod(683(3))].">
      <j1 rule="back_demod" parents="94"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[back_demod(80),demod(683(4),683(3))].">
      <j1 rule="back_demod" parents="80"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="1016">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ (z v (x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(683(a,1),746(a,1,1,2,1)),demod(683(10))].">
      <j1 rule="para" parents="683 746"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="1454">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ (z v (((x v y) ^ y) v ((x v y) ^ y))) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(44(a,1),1016(a,1,1,2))].">
      <j1 rule="para" parents="44 1016"/>
    </justification>
  </clause>

  <clause id="1649">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),748(a,1,1,2,1,2))].">
      <j1 rule="para" parents="652 748"/>
    </justification>
  </clause>

  <clause id="1705">
    <literal><![CDATA[
      (x v x) ^ (y v ((x v x) ^ ((x v x) v z))) = x v x
    ]]></literal>
    <justification jstring="[para(683(a,1),797(a,1,2,2,1))].">
      <j1 rule="para" parents="683 797"/>
    </justification>
  </clause>

  <clause id="1707">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v ((u v (x v x)) ^ (x v x))) ^ (((u v (x v x)) ^ (x v x)) v v))) = x v x
    ]]></literal>
    <justification jstring="[para(30(a,1),797(a,1,2,2))].">
      <j1 rule="para" parents="30 797"/>
    </justification>
  </clause>

  <clause id="1745">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(44(a,1),1705(a,1,2,2)),demod(657(7)),flip(a)].">
      <j1 rule="para" parents="44 1705"/>
      <j2 rule="demod" parents="657"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1794">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_demod(1454),demod(1745(5),1745(6),792(3),1745(6))].">
      <j1 rule="back_demod" parents="1454"/>
      <j2 rule="demod" parents="1745 1745 792 1745"/>
    </justification>
  </clause>

  <clause id="1822">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_demod(1745),demod(1794(5))].">
      <j1 rule="back_demod" parents="1745"/>
      <j2 rule="demod" parents="1794"/>
    </justification>
  </clause>

  <clause id="1840">
    <literal><![CDATA[
      x ^ (y v ((z v x) ^ (x v u))) = x
    ]]></literal>
    <justification jstring="[back_demod(1707),demod(1794(1),1794(1),1794(2),1822(2),1794(2),1794(3),1822(3),1794(6))].">
      <j1 rule="back_demod" parents="1707"/>
      <j2 rule="demod" parents="1794 1794 1794 1822 1794 1794 1822 1794"/>
    </justification>
  </clause>

  <clause id="1841">
    <literal><![CDATA[
      x ^ (y v (x ^ (x v z))) = x
    ]]></literal>
    <justification jstring="[back_demod(1705),demod(1794(1),1794(1),1794(1),1794(5))].">
      <j1 rule="back_demod" parents="1705"/>
      <j2 rule="demod" parents="1794 1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="1883">
    <literal><![CDATA[
      (x v (x ^ y)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_demod(1649),demod(1794(1),1794(1),1794(5))].">
      <j1 rule="back_demod" parents="1649"/>
      <j2 rule="demod" parents="1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="2202">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_demod(689),demod(1794(5))].">
      <j1 rule="back_demod" parents="689"/>
      <j2 rule="demod" parents="1794"/>
    </justification>
  </clause>

  <clause id="2334">
    <literal><![CDATA[
      x ^ (x ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[para(1794(a,1),1841(a,1,2))].">
      <j1 rule="para" parents="1794 1841"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(1883(a,1),1841(a,1,2,2)),demod(1883(4)),flip(a)].">
      <j1 rule="para" parents="1883 1841"/>
      <j2 rule="demod" parents="1883"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2341">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(2340(a,1),1822(a,1,1))].">
      <j1 rule="para" parents="2340 1822"/>
    </justification>
  </clause>

  <clause id="2345">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_demod(2334),demod(2341(3))].">
      <j1 rule="back_demod" parents="2334"/>
      <j2 rule="demod" parents="2341"/>
    </justification>
  </clause>

  <clause id="2376">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(2202(a,1),2345(a,1)),flip(a)].">
      <j1 rule="para" parents="2202 2345"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2443">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(2376(a,1),1822(a,1,1))].">
      <j1 rule="para" parents="2376 1822"/>
    </justification>
  </clause>

  <clause id="2751">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[para(1794(a,1),1840(a,1,2))].">
      <j1 rule="para" parents="1794 1840"/>
    </justification>
  </clause>

  <clause id="3223">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(2751(a,1),2443(a,1,2)),demod(2751(8))].">
      <j1 rule="para" parents="2751 2443"/>
      <j2 rule="demod" parents="2751"/>
    </justification>
  </clause>

  <clause id="3224">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_4)
    ]]></attribute>
    <justification jstring="[resolve(3223,a,10,a)].">
      <j1 rule="resolve" parents="3223 10"/>
    </justification>
  </clause>

</proof>

<proof number="4" length="75" max_count="89">

<comments><![CDATA[
% Proof 4 at 4.31 (+ 0.06) seconds: McKenzie_3.
% Length of proof is 75.
% Level of proof is 31.
% Maximum clause weight is 89.
% Given clauses 353.
]]></comments>

  <clause id="6">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v ((v6 v y) ^ (y v v7))) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A1)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="9">
    <literal><![CDATA[
      ((c7 ^ c8) v (c8 ^ c9)) v c8 != c8
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_3)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (((x v ((y v z) ^ (z v u))) ^ ((y v z) ^ (z v u))) v ((z v (v ^ ((y v z) ^ (z v u)))) ^ w)) ^ (v6 v ((v7 v ((y v z) ^ (z v u))) ^ (((y v z) ^ (z v u)) v v8))) = (y v z) ^ (z v u)
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1,2,1,1))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,1,2))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      (((x v (((y ^ (z v z)) v (u ^ z)) ^ v)) ^ (((y ^ (z v z)) v (u ^ z)) ^ v)) v (((w ^ ((((y ^ (z v z)) v (u ^ z)) ^ v) v (((y ^ (z v z)) v (u ^ z)) ^ v))) v (v6 ^ (((y ^ (z v z)) v (u ^ z)) ^ v))) ^ v7)) ^ (v8 v z) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="13 13"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (v v ((w v y) ^ (y v v6))) = y
    ]]></literal>
    <justification jstring="[para(17(a,1),6(a,1,1,2,1,1))].">
      <j1 rule="para" parents="17 6"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (v v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),18(a,1,2,2))].">
      <j1 rule="para" parents="13 18"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(17(a,1),24(a,1,1,2,1,2))].">
      <j1 rule="para" parents="17 24"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      (((x v ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) ^ ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) v u) ^ (w v ((v6 v ((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v))) ^ (((y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v)) v v7))) = (y v ((z v u) ^ u)) ^ (((z v u) ^ u) v v)
    ]]></literal>
    <justification jstring="[para(6(a,1),11(a,1,1,2))].">
      <j1 rule="para" parents="6 11"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (v6 v ((v7 v y) ^ (y v v8))) = y
    ]]></literal>
    <justification jstring="[para(18(a,1),11(a,1,1,1,1,2)),demod(18(17),18(22),18(26),18(27),18(31))].">
      <j1 rule="para" parents="18 11"/>
      <j2 rule="demod" parents="18 18 18 18 18"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v ((v v y) ^ (y v w))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),34(a,1,1,2))].">
      <j1 rule="para" parents="6 34"/>
    </justification>
  </clause>

  <clause id="48">
    <literal><![CDATA[
      (((x v (y v z)) ^ (y v z)) v ((((((y v z) v (y v z)) v (z v z)) ^ u) v (v ^ (y v z))) ^ w)) ^ (v6 v ((v7 v (y v z)) ^ ((y v z) v v8))) = y v z
    ]]></literal>
    <justification jstring="[para(17(a,1),34(a,1,1,2,1,1,1,2))].">
      <j1 rule="para" parents="17 34"/>
    </justification>
  </clause>

  <clause id="57">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(13(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="13 44"/>
    </justification>
  </clause>

  <clause id="67">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (v v z) = y ^ z
    ]]></literal>
    <justification jstring="[para(44(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="44 44"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (z v y) = y v y
    ]]></literal>
    <justification jstring="[para(17(a,1),57(a,1,1,2))].">
      <j1 rule="para" parents="17 57"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),13(a,1,1,1))].">
      <j1 rule="para" parents="69 13"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),57(a,1,1,1))].">
      <j1 rule="para" parents="69 57"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),69(a,1,1,1))].">
      <j1 rule="para" parents="69 69"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(14(a,1),92(a,1,1,2))].">
      <j1 rule="para" parents="14 92"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v (v v ((w v y) ^ (y v v6)))) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),67(a,1,1,1,1,2)),demod(6(14),6(15),6(23))].">
      <j1 rule="para" parents="6 67"/>
      <j2 rule="demod" parents="6 6 6"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ y)) ^ (u v y) = y v y
    ]]></literal>
    <justification jstring="[para(67(a,1),57(a,1,1,2))].">
      <j1 rule="para" parents="67 57"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (v v x) = x v x
    ]]></literal>
    <justification jstring="[para(67(a,1),103(a,1,1,2,1,1))].">
      <j1 rule="para" parents="67 103"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      ((x v x) v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(6(a,1),148(a,1,1,2))].">
      <j1 rule="para" parents="6 148"/>
    </justification>
  </clause>

  <clause id="185">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(171(a,1),6(a,1,2,2))].">
      <j1 rule="para" parents="171 6"/>
    </justification>
  </clause>

  <clause id="605">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ (w v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(67(a,1),185(a,1,1,2,1,1))].">
      <j1 rule="para" parents="67 185"/>
    </justification>
  </clause>

  <clause id="624">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(6(a,1),605(a,1,1,2))].">
      <j1 rule="para" parents="6 605"/>
    </justification>
  </clause>

  <clause id="652">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (z v y) = y
    ]]></literal>
    <justification jstring="[para(624(a,1),13(a,1,2,2))].">
      <j1 rule="para" parents="624 13"/>
    </justification>
  </clause>

  <clause id="657">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (u v y) = y
    ]]></literal>
    <justification jstring="[para(624(a,1),44(a,1,2,2))].">
      <j1 rule="para" parents="624 44"/>
    </justification>
  </clause>

  <clause id="670">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),24(a,1,1,1))].">
      <j1 rule="para" parents="652 24"/>
    </justification>
  </clause>

  <clause id="672">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),28(a,1,1,1))].">
      <j1 rule="para" parents="652 28"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v (x v x)) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),69(a,1,1,1))].">
      <j1 rule="para" parents="652 69"/>
    </justification>
  </clause>

  <clause id="683">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(69(a,1),652(a,1,1,1)),demod(682(8))].">
      <j1 rule="para" parents="69 652"/>
      <j2 rule="demod" parents="682"/>
    </justification>
  </clause>

  <clause id="689">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),112(a,1,1,1))].">
      <j1 rule="para" parents="652 112"/>
    </justification>
  </clause>

  <clause id="739">
    <literal><![CDATA[
      (((x v y) ^ y) v ((y v (z ^ y)) ^ u)) ^ (v v (y v y)) = y
    ]]></literal>
    <justification jstring="[para(652(a,1),185(a,1,1,2,1,1))].">
      <j1 rule="para" parents="652 185"/>
    </justification>
  </clause>

  <clause id="746">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(672),demod(683(3),683(3))].">
      <j1 rule="back_demod" parents="672"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="748">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (u v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(670),demod(683(3))].">
      <j1 rule="back_demod" parents="670"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="792">
    <literal><![CDATA[
      (x v x) ^ (y v x) = x v x
    ]]></literal>
    <justification jstring="[back_demod(94),demod(683(3))].">
      <j1 rule="back_demod" parents="94"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      (x v x) ^ (y v ((z v (x v x)) ^ ((x v x) v u))) = x v x
    ]]></literal>
    <justification jstring="[back_demod(80),demod(683(4),683(3))].">
      <j1 rule="back_demod" parents="80"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="920">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (z v ((u v y) ^ (y v v))) = y
    ]]></literal>
    <justification jstring="[para(44(a,1),30(a,1,1,1,1,2)),demod(44(13),44(15),44(16),44(20))].">
      <j1 rule="para" parents="44 30"/>
      <j2 rule="demod" parents="44 44 44 44"/>
    </justification>
  </clause>

  <clause id="1016">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ (z v (x v x)) = x v x
    ]]></literal>
    <justification jstring="[para(683(a,1),746(a,1,1,2,1)),demod(683(10))].">
      <j1 rule="para" parents="683 746"/>
      <j2 rule="demod" parents="683"/>
    </justification>
  </clause>

  <clause id="1454">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ (z v (((x v y) ^ y) v ((x v y) ^ y))) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(44(a,1),1016(a,1,1,2))].">
      <j1 rule="para" parents="44 1016"/>
    </justification>
  </clause>

  <clause id="1605">
    <literal><![CDATA[
      (((x v y) ^ y) v ((y v (z ^ y)) ^ u)) ^ (y v y) = y
    ]]></literal>
    <justification jstring="[para(683(a,1),739(a,1,2))].">
      <j1 rule="para" parents="683 739"/>
    </justification>
  </clause>

  <clause id="1649">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (z v x) = x v x
    ]]></literal>
    <justification jstring="[para(652(a,1),748(a,1,1,2,1,2))].">
      <j1 rule="para" parents="652 748"/>
    </justification>
  </clause>

  <clause id="1663">
    <literal><![CDATA[
      (((x v (((y v y) v y) ^ z)) ^ (((y v y) v y) ^ z)) v (((y v y) v y) ^ z)) ^ (u v (y v y)) = ((y v y) v y) ^ z
    ]]></literal>
    <justification jstring="[para(1649(a,1),920(a,1,2,2))].">
      <j1 rule="para" parents="1649 920"/>
    </justification>
  </clause>

  <clause id="1686">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) ^ z) v (u ^ (y v y))) ^ v)) ^ (w v ((v6 v (y v y)) ^ (y v y))) = y v y
    ]]></literal>
    <justification jstring="[para(683(a,1),48(a,1,2,2,2)),demod(683(7),683(7))].">
      <j1 rule="para" parents="683 48"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="1703">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) ^ z) v y) ^ u)) ^ (v v ((w v (y v y)) ^ ((y v y) v v6))) = y v y
    ]]></literal>
    <justification jstring="[para(1605(a,1),48(a,1,1,2,1,2)),demod(683(7),683(7))].">
      <j1 rule="para" parents="1605 48"/>
      <j2 rule="demod" parents="683 683"/>
    </justification>
  </clause>

  <clause id="1705">
    <literal><![CDATA[
      (x v x) ^ (y v ((x v x) ^ ((x v x) v z))) = x v x
    ]]></literal>
    <justification jstring="[para(683(a,1),797(a,1,2,2,1))].">
      <j1 rule="para" parents="683 797"/>
    </justification>
  </clause>

  <clause id="1745">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(44(a,1),1705(a,1,2,2)),demod(657(7)),flip(a)].">
      <j1 rule="para" parents="44 1705"/>
      <j2 rule="demod" parents="657"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1757">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(105(a,1),1705(a,1,2,2)),demod(1745(5),1745(7))].">
      <j1 rule="para" parents="105 1705"/>
      <j2 rule="demod" parents="1745 1745"/>
    </justification>
  </clause>

  <clause id="1794">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_demod(1454),demod(1745(5),1745(6),792(3),1745(6))].">
      <j1 rule="back_demod" parents="1454"/>
      <j2 rule="demod" parents="1745 1745 792 1745"/>
    </justification>
  </clause>

  <clause id="1822">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_demod(1745),demod(1794(5))].">
      <j1 rule="back_demod" parents="1745"/>
      <j2 rule="demod" parents="1794"/>
    </justification>
  </clause>

  <clause id="1841">
    <literal><![CDATA[
      x ^ (y v (x ^ (x v z))) = x
    ]]></literal>
    <justification jstring="[back_demod(1705),demod(1794(1),1794(1),1794(1),1794(5))].">
      <j1 rule="back_demod" parents="1705"/>
      <j2 rule="demod" parents="1794 1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="1843">
    <literal><![CDATA[
      (x v (((x ^ y) v x) ^ z)) ^ (u v ((v v x) ^ (x v w))) = x
    ]]></literal>
    <justification jstring="[back_demod(1703),demod(1794(1),1794(2),1822(2),1794(1),1794(5),1794(6),1794(10))].">
      <j1 rule="back_demod" parents="1703"/>
      <j2 rule="demod" parents="1794 1794 1822 1794 1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="1856">
    <literal><![CDATA[
      (x v (((x ^ y) v (z ^ x)) ^ u)) ^ (v v x) = x
    ]]></literal>
    <justification jstring="[back_demod(1686),demod(1794(1),1794(2),1822(2),1794(1),1794(2),1794(6),1794(7),1822(7),1794(8))].">
      <j1 rule="back_demod" parents="1686"/>
      <j2 rule="demod" parents="1794 1794 1822 1794 1794 1794 1794 1822 1794"/>
    </justification>
  </clause>

  <clause id="1871">
    <literal><![CDATA[
      (x ^ y) ^ (z v x) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(1663),demod(1794(1),1794(1),1794(3),1794(3),1822(4),1794(2),1794(2),1794(3),1794(2),1794(4),1794(4))].">
      <j1 rule="back_demod" parents="1663"/>
      <j2 rule="demod" parents="1794 1794 1794 1794 1822 1794 1794 1794 1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="1883">
    <literal><![CDATA[
      (x v (x ^ y)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_demod(1649),demod(1794(1),1794(1),1794(5))].">
      <j1 rule="back_demod" parents="1649"/>
      <j2 rule="demod" parents="1794 1794 1794"/>
    </justification>
  </clause>

  <clause id="2202">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (z v x) = x
    ]]></literal>
    <justification jstring="[back_demod(689),demod(1794(5))].">
      <j1 rule="back_demod" parents="689"/>
      <j2 rule="demod" parents="1794"/>
    </justification>
  </clause>

  <clause id="2334">
    <literal><![CDATA[
      x ^ (x ^ (x v y)) = x
    ]]></literal>
    <justification jstring="[para(1794(a,1),1841(a,1,2))].">
      <j1 rule="para" parents="1794 1841"/>
    </justification>
  </clause>

  <clause id="2336">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(1794(a,1),1871(a,1,2))].">
      <j1 rule="para" parents="1794 1871"/>
    </justification>
  </clause>

  <clause id="2340">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[para(1883(a,1),1841(a,1,2,2)),demod(1883(4)),flip(a)].">
      <j1 rule="para" parents="1883 1841"/>
      <j2 rule="demod" parents="1883"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2341">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(2340(a,1),1822(a,1,1))].">
      <j1 rule="para" parents="2340 1822"/>
    </justification>
  </clause>

  <clause id="2345">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[back_demod(2334),demod(2341(3))].">
      <j1 rule="back_demod" parents="2334"/>
      <j2 rule="demod" parents="2341"/>
    </justification>
  </clause>

  <clause id="2376">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(2202(a,1),2345(a,1)),flip(a)].">
      <j1 rule="para" parents="2202 2345"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2443">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(2376(a,1),1822(a,1,1))].">
      <j1 rule="para" parents="2376 1822"/>
    </justification>
  </clause>

  <clause id="2824">
    <literal><![CDATA[
      x v (((x ^ y) v x) ^ z) = x
    ]]></literal>
    <justification jstring="[para(1843(a,1),2345(a,1)),flip(a)].">
      <j1 rule="para" parents="1843 2345"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3102">
    <literal><![CDATA[
      x v (((x ^ y) v (z ^ x)) ^ u) = x
    ]]></literal>
    <justification jstring="[para(1856(a,1),2345(a,1)),flip(a)].">
      <j1 rule="para" parents="1856 2345"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3252">
    <literal><![CDATA[
      x v ((x ^ y) v x) = x
    ]]></literal>
    <justification jstring="[para(1757(a,1),2824(a,1,2))].">
      <j1 rule="para" parents="1757 2824"/>
    </justification>
  </clause>

  <clause id="3292">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(3252(a,1),1757(a,1,2)),demod(1822(3)),flip(a)].">
      <j1 rule="para" parents="3252 1757"/>
      <j2 rule="demod" parents="1822"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3312">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1822(a,1),3292(a,1,1))].">
      <j1 rule="para" parents="1822 3292"/>
    </justification>
  </clause>

  <clause id="9078">
    <literal><![CDATA[
      x v ((x ^ y) v (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(1757(a,1),3102(a,1,2))].">
      <j1 rule="para" parents="1757 3102"/>
    </justification>
  </clause>

  <clause id="9236">
    <literal><![CDATA[
      x v ((x ^ y) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2336(a,1),9078(a,1,2,2))].">
      <j1 rule="para" parents="2336 9078"/>
    </justification>
  </clause>

  <clause id="9396">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2443(a,1),9236(a,1,2,1))].">
      <j1 rule="para" parents="2443 9236"/>
    </justification>
  </clause>

  <clause id="9827">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[para(9396(a,1),3312(a,1,2)),demod(9396(8))].">
      <j1 rule="para" parents="9396 3312"/>
      <j2 rule="demod" parents="9396"/>
    </justification>
  </clause>

  <clause id="9828">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie_3)
    ]]></attribute>
    <justification jstring="[resolve(9827,a,9,a)].">
      <j1 rule="resolve" parents="9827 9"/>
    </justification>
  </clause>

</proof>

</proofs>
