<?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>a2.out</source>

<heading><![CDATA[
Prover9 (32) version November-2006, November 2006.
Process 3830 was started by mccune on cleo.thornwood,
Wed Nov 22 11:23:58 2006
The command was "/home/mccune/bin/prover9 -f a2.in".
]]></heading>

<proof number="1" length="101" max_count="89">

<comments><![CDATA[
% Proof 1 at 4.26 (+ 0.05) seconds: McKenzie.
% Length of proof is 101.
% Level of proof is 32.
% Maximum clause weight is 89.
% Given clauses 350.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x & x ^ (y v (x v z)) = x & ((y ^ x) v (x ^ z)) v x = x & ((y v x) ^ (x v z)) ^ x = x
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <attribute><![CDATA[
      label(goal)
    ]]></attribute>
    <justification jstring="[goal].">
      <j1 rule="goal"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ (((w v y) ^ (v6 v y)) v v7) = y
    ]]></literal>
    <attribute><![CDATA[
      label(A2)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="deny">
    <literal><![CDATA[
      c1 v (c2 ^ (c1 ^ c3)) != c1
    ]]></literal>
    <literal><![CDATA[
      c1 ^ (c2 v (c1 v c3)) != c1
    ]]></literal>
    <literal><![CDATA[
      ((c2 ^ c1) v (c1 ^ c3)) v c1 != c1
    ]]></literal>
    <literal><![CDATA[
      ((c2 v c1) ^ (c1 v c3)) ^ c1 != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      (((x v ((y v z) ^ (u v z))) ^ ((y v z) ^ (u v z))) v ((z v (v ^ ((y v z) ^ (u v z)))) ^ w)) ^ (((v6 v ((y v z) ^ (u v z))) ^ (v7 v ((y v z) ^ (u v z)))) v v8) = (y v z) ^ (u v z)
    ]]></literal>
    <justification jstring="[para(2(a,1),2(a,1,1,2,1,1))].">
      <j1 rule="para" parents="2 2"/>
    </justification>
  </clause>

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

  <clause id="7">
    <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)) ^ (z v v8) = ((y ^ (z v z)) v (u ^ z)) ^ v
    ]]></literal>
    <justification jstring="[para(2(a,1),2(a,1,2,1))].">
      <j1 rule="para" parents="2 2"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((y v y) v (y v y))) ^ (y v z) = y v y
    ]]></literal>
    <justification jstring="[para(6(a,1),6(a,1,2,1))].">
      <j1 rule="para" parents="6 6"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      (((x v y) ^ y) v (((y v y) v (z ^ y)) ^ u)) ^ (((v v y) ^ (w v y)) v v6) = y
    ]]></literal>
    <justification jstring="[para(10(a,1),2(a,1,1,2,1,1))].">
      <j1 rule="para" parents="10 2"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (z ^ (y v y))) ^ u)) ^ (y v v) = y v y
    ]]></literal>
    <justification jstring="[para(6(a,1),11(a,1,2,1))].">
      <j1 rule="para" parents="6 11"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (((x v ((y v ((z v u) ^ u)) ^ (v v ((z v u) ^ u)))) ^ ((y v ((z v u) ^ u)) ^ (v v ((z v u) ^ u)))) v u) ^ (((w v ((y v ((z v u) ^ u)) ^ (v v ((z v u) ^ u)))) ^ (v6 v ((y v ((z v u) ^ u)) ^ (v v ((z v u) ^ u))))) v v7) = (y v ((z v u) ^ u)) ^ (v v ((z v u) ^ u))
    ]]></literal>
    <justification jstring="[para(2(a,1),4(a,1,1,2))].">
      <j1 rule="para" parents="2 4"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      (((x v y) ^ y) v (((((y v y) v (z ^ y)) ^ u) v (v ^ y)) ^ w)) ^ (((v6 v y) ^ (v7 v y)) v v8) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),4(a,1,1,1,1,2)),rewrite(11(17),11(22),11(26),11(27),11(31))].">
      <j1 rule="para" parents="11 4"/>
      <j2 rule="rewrite" parents="11 11 11 11 11"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) v (y v y)) v (y v y)) ^ z)) ^ (y v u) = y v y
    ]]></literal>
    <justification jstring="[para(10(a,1),17(a,1,1,2,1,2))].">
      <j1 rule="para" parents="10 17"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (((u v y) ^ (v v y)) v w) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),25(a,1,1,2))].">
      <j1 rule="para" parents="2 25"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (((x v (y v z)) ^ (y v z)) v ((((((y v z) v (y v z)) v (y v y)) ^ u) v (v ^ (y v z))) ^ w)) ^ (((v6 v (y v z)) ^ (v7 v (y v z))) v v8) = y v z
    ]]></literal>
    <justification jstring="[para(10(a,1),25(a,1,1,2,1,1,1,2))].">
      <j1 rule="para" parents="10 25"/>
    </justification>
  </clause>

  <clause id="50">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ (y v y))) ^ (y v u) = y v y
    ]]></literal>
    <justification jstring="[para(6(a,1),37(a,1,2,1))].">
      <j1 rule="para" parents="6 37"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (z v v) = y ^ z
    ]]></literal>
    <justification jstring="[para(37(a,1),37(a,1,2,1))].">
      <j1 rule="para" parents="37 37"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (y v z) = y v y
    ]]></literal>
    <justification jstring="[para(10(a,1),50(a,1,1,2))].">
      <j1 rule="para" parents="10 50"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      ((x v x) v ((x v x) v (x v x))) ^ (((y v (x v x)) ^ (z v (x v x))) v u) = x v x
    ]]></literal>
    <justification jstring="[para(70(a,1),6(a,1,1,1))].">
      <j1 rule="para" parents="70 6"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(70(a,1),50(a,1,1,1))].">
      <j1 rule="para" parents="70 50"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (x v y) = x v x
    ]]></literal>
    <justification jstring="[para(70(a,1),70(a,1,1,1))].">
      <j1 rule="para" parents="70 70"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (x v v) = x v x
    ]]></literal>
    <justification jstring="[para(7(a,1),86(a,1,1,2))].">
      <j1 rule="para" parents="7 86"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ ((((u v y) ^ (v v y)) v w) v v6) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),60(a,1,1,1,1,2)),rewrite(2(14),2(15),2(23))].">
      <j1 rule="para" parents="2 60"/>
      <j2 rule="rewrite" parents="2 2 2"/>
    </justification>
  </clause>

  <clause id="106">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ y)) ^ (y v u) = y v y
    ]]></literal>
    <justification jstring="[para(60(a,1),50(a,1,1,2))].">
      <j1 rule="para" parents="60 50"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      ((x v x) v (y ^ x)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(60(a,1),86(a,1,1,2))].">
      <j1 rule="para" parents="60 86"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      (((x v y) v (x v y)) v (x v x)) ^ ((x v y) v z) = (x v y) v (x v y)
    ]]></literal>
    <justification jstring="[para(10(a,1),107(a,1,1,2))].">
      <j1 rule="para" parents="10 107"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (x v v) = x v x
    ]]></literal>
    <justification jstring="[para(60(a,1),96(a,1,1,2,1,1))].">
      <j1 rule="para" parents="60 96"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      ((x v x) v x) ^ (x v y) = x v x
    ]]></literal>
    <justification jstring="[para(2(a,1),147(a,1,1,2))].">
      <j1 rule="para" parents="2 147"/>
    </justification>
  </clause>

  <clause id="178">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ (y v y)) v (u ^ y)) ^ v)) ^ ((y v y) v w) = y
    ]]></literal>
    <justification jstring="[para(164(a,1),2(a,1,2,1))].">
      <j1 rule="para" parents="164 2"/>
    </justification>
  </clause>

  <clause id="179">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ ((y v y) v z) = y
    ]]></literal>
    <justification jstring="[para(164(a,1),6(a,1,2,1))].">
      <j1 rule="para" parents="164 6"/>
    </justification>
  </clause>

  <clause id="201">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (((y v y) v u) v v) = y
    ]]></literal>
    <justification jstring="[para(179(a,1),60(a,1,1,1,1,2)),rewrite(179(8),179(9),179(15))].">
      <j1 rule="para" parents="179 60"/>
      <j2 rule="rewrite" parents="179 179 179"/>
    </justification>
  </clause>

  <clause id="231">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ ((((y v y) v u) v v) v w) = y
    ]]></literal>
    <justification jstring="[para(201(a,1),60(a,1,1,1,1,2)),rewrite(201(9),201(10),201(17))].">
      <j1 rule="para" parents="201 60"/>
      <j2 rule="rewrite" parents="201 201 201"/>
    </justification>
  </clause>

  <clause id="599">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ ((y v y) v w) = y
    ]]></literal>
    <justification jstring="[para(60(a,1),178(a,1,1,2,1,1))].">
      <j1 rule="para" parents="60 178"/>
    </justification>
  </clause>

  <clause id="617">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ ((y v y) v z) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),599(a,1,1,2))].">
      <j1 rule="para" parents="2 599"/>
    </justification>
  </clause>

  <clause id="645">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (y v z) = y
    ]]></literal>
    <justification jstring="[para(617(a,1),6(a,1,2,1))].">
      <j1 rule="para" parents="617 6"/>
    </justification>
  </clause>

  <clause id="650">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (y v u) = y
    ]]></literal>
    <justification jstring="[para(617(a,1),37(a,1,2,1))].">
      <j1 rule="para" parents="617 37"/>
    </justification>
  </clause>

  <clause id="663">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (y ^ (x v x))) ^ z)) ^ (x v u) = x v x
    ]]></literal>
    <justification jstring="[para(645(a,1),17(a,1,1,1))].">
      <j1 rule="para" parents="645 17"/>
    </justification>
  </clause>

  <clause id="665">
    <literal><![CDATA[
      (x v ((((x v x) v (x v x)) v (x v x)) ^ y)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(645(a,1),29(a,1,1,1))].">
      <j1 rule="para" parents="645 29"/>
    </justification>
  </clause>

  <clause id="675">
    <literal><![CDATA[
      (x v (x v x)) ^ (x v y) = x v x
    ]]></literal>
    <justification jstring="[para(645(a,1),70(a,1,1,1))].">
      <j1 rule="para" parents="645 70"/>
    </justification>
  </clause>

  <clause id="676">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(70(a,1),645(a,1,1,1)),rewrite(675(8))].">
      <j1 rule="para" parents="70 645"/>
      <j2 rule="rewrite" parents="675"/>
    </justification>
  </clause>

  <clause id="682">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(645(a,1),106(a,1,1,1))].">
      <j1 rule="para" parents="645 106"/>
    </justification>
  </clause>

  <clause id="695">
    <literal><![CDATA[
      (x v (y ^ (x v x))) ^ ((((x v x) v z) v u) v v) = x v x
    ]]></literal>
    <justification jstring="[para(645(a,1),231(a,1,1,1)),rewrite(676(6))].">
      <j1 rule="para" parents="645 231"/>
      <j2 rule="rewrite" parents="676"/>
    </justification>
  </clause>

  <clause id="733">
    <literal><![CDATA[
      (((x v y) ^ y) v ((y v (z ^ y)) ^ u)) ^ ((y v y) v v) = y
    ]]></literal>
    <justification jstring="[para(645(a,1),178(a,1,1,2,1,1))].">
      <j1 rule="para" parents="645 178"/>
    </justification>
  </clause>

  <clause id="739">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(665),rewrite(676(3),676(3))].">
      <j1 rule="back_rewrite" parents="665"/>
      <j2 rule="rewrite" parents="676 676"/>
    </justification>
  </clause>

  <clause id="741">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (x v u) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(663),rewrite(676(3))].">
      <j1 rule="back_rewrite" parents="663"/>
      <j2 rule="rewrite" parents="676"/>
    </justification>
  </clause>

  <clause id="785">
    <literal><![CDATA[
      (x v x) ^ (x v y) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(87),rewrite(676(3))].">
      <j1 rule="back_rewrite" parents="87"/>
      <j2 rule="rewrite" parents="676"/>
    </justification>
  </clause>

  <clause id="790">
    <literal><![CDATA[
      (x v x) ^ (((y v (x v x)) ^ (z v (x v x))) v u) = x v x
    ]]></literal>
    <justification jstring="[back_rewrite(73),rewrite(676(4),676(3))].">
      <j1 rule="back_rewrite" parents="73"/>
      <j2 rule="rewrite" parents="676 676"/>
    </justification>
  </clause>

  <clause id="882">
    <literal><![CDATA[
      ((x v x) v ((y ^ x) v (y ^ x))) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(785(a,1),147(a,1,1,2))].">
      <j1 rule="para" parents="785 147"/>
    </justification>
  </clause>

  <clause id="912">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (((z v y) ^ (u v y)) v v) = y
    ]]></literal>
    <justification jstring="[para(37(a,1),21(a,1,1,1,1,2)),rewrite(37(13),37(15),37(16),37(20))].">
      <j1 rule="para" parents="37 21"/>
      <j2 rule="rewrite" parents="37 37 37 37"/>
    </justification>
  </clause>

  <clause id="1011">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ ((x v x) v z) = x v x
    ]]></literal>
    <justification jstring="[para(676(a,1),739(a,1,1,2,1)),rewrite(676(10))].">
      <j1 rule="para" parents="676 739"/>
      <j2 rule="rewrite" parents="676"/>
    </justification>
  </clause>

  <clause id="1446">
    <literal><![CDATA[
      ((((x v y) ^ y) v ((x v y) ^ y)) v y) ^ ((((x v y) ^ y) v ((x v y) ^ y)) v z) = ((x v y) ^ y) v ((x v y) ^ y)
    ]]></literal>
    <justification jstring="[para(37(a,1),1011(a,1,1,2))].">
      <j1 rule="para" parents="37 1011"/>
    </justification>
  </clause>

  <clause id="1513">
    <literal><![CDATA[
      (x v (y ^ x)) ^ ((((x v x) v z) v u) v v) = x v x
    ]]></literal>
    <justification jstring="[para(60(a,1),695(a,1,1,2))].">
      <j1 rule="para" parents="60 695"/>
    </justification>
  </clause>

  <clause id="1598">
    <literal><![CDATA[
      (((x v y) ^ y) v ((y v (z ^ y)) ^ u)) ^ (y v y) = y
    ]]></literal>
    <justification jstring="[para(676(a,1),733(a,1,2))].">
      <j1 rule="para" parents="676 733"/>
    </justification>
  </clause>

  <clause id="1615">
    <literal><![CDATA[
      (((x v ((y v z) ^ z)) ^ ((y v z) ^ z)) v z) ^ (((y v z) ^ z) v ((y v z) ^ z)) = (y v z) ^ z
    ]]></literal>
    <justification jstring="[para(2(a,1),1598(a,1,1,2))].">
      <j1 rule="para" parents="2 1598"/>
    </justification>
  </clause>

  <clause id="1641">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(645(a,1),741(a,1,1,2,1,2))].">
      <j1 rule="para" parents="645 741"/>
    </justification>
  </clause>

  <clause id="1664">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) ^ z) v (u ^ (y v y))) ^ v)) ^ (((w v (y v y)) ^ (y v y)) v v6) = y v y
    ]]></literal>
    <justification jstring="[para(676(a,1),41(a,1,2,1,2)),rewrite(676(7),676(7))].">
      <j1 rule="para" parents="676 41"/>
      <j2 rule="rewrite" parents="676 676"/>
    </justification>
  </clause>

  <clause id="1681">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v ((((y v y) ^ z) v y) ^ u)) ^ (((v v (y v y)) ^ (w v (y v y))) v v6) = y v y
    ]]></literal>
    <justification jstring="[para(1598(a,1),41(a,1,1,2,1,2)),rewrite(676(7),676(7))].">
      <j1 rule="para" parents="1598 41"/>
      <j2 rule="rewrite" parents="676 676"/>
    </justification>
  </clause>

  <clause id="1694">
    <literal><![CDATA[
      (((x v (((y v y) v y) ^ z)) ^ (((y v y) v y) ^ z)) v (((y v y) v y) ^ z)) ^ ((y v y) v u) = ((y v y) v y) ^ z
    ]]></literal>
    <justification jstring="[para(1641(a,1),912(a,1,2,1))].">
      <j1 rule="para" parents="1641 912"/>
    </justification>
  </clause>

  <clause id="1696">
    <literal><![CDATA[
      (x v x) ^ (((x v y) v (x v y)) v z) = x v x
    ]]></literal>
    <justification jstring="[para(115(a,1),790(a,1,2,1))].">
      <j1 rule="para" parents="115 790"/>
    </justification>
  </clause>

  <clause id="1697">
    <literal><![CDATA[
      (x v x) ^ (((x v x) ^ (y v (x v x))) v z) = x v x
    ]]></literal>
    <justification jstring="[para(676(a,1),790(a,1,2,1,1))].">
      <j1 rule="para" parents="676 790"/>
    </justification>
  </clause>

  <clause id="1699">
    <literal><![CDATA[
      (x v x) ^ (((y v ((z v (x v x)) ^ (x v x))) ^ (u v ((z v (x v x)) ^ (x v x)))) v v) = x v x
    ]]></literal>
    <justification jstring="[para(21(a,1),790(a,1,2,1))].">
      <j1 rule="para" parents="21 790"/>
    </justification>
  </clause>

  <clause id="1700">
    <literal><![CDATA[
      ((x ^ y) v (x ^ y)) ^ ((y v y) v z) = (x ^ y) v (x ^ y)
    ]]></literal>
    <justification jstring="[para(882(a,1),790(a,1,2,1))].">
      <j1 rule="para" parents="882 790"/>
    </justification>
  </clause>

  <clause id="1737">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(37(a,1),1697(a,1,2,1)),rewrite(650(7)),flip(a)].">
      <j1 rule="para" parents="37 1697"/>
      <j2 rule="rewrite" parents="650"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1749">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(98(a,1),1697(a,1,2,1)),rewrite(1737(5),1737(7))].">
      <j1 rule="para" parents="98 1697"/>
      <j2 rule="rewrite" parents="1737 1737"/>
    </justification>
  </clause>

  <clause id="1780">
    <literal><![CDATA[
      ((((x v x) ^ (y v (x v x))) v z) v (x v x)) ^ (((((((x v x) ^ (y v (x v x))) v z) v (((x v x) ^ (y v (x v x))) v z)) v u) v v) v w) = (((x v x) ^ (y v (x v x))) v z) v (((x v x) ^ (y v (x v x))) v z)
    ]]></literal>
    <justification jstring="[para(1697(a,1),1513(a,1,1,2))].">
      <j1 rule="para" parents="1697 1513"/>
    </justification>
  </clause>

  <clause id="1784">
    <literal><![CDATA[
      (((x v ((y v z) ^ z)) ^ ((y v z) ^ z)) v z) ^ z = (y v z) ^ z
    ]]></literal>
    <justification jstring="[back_rewrite(1615),rewrite(1737(12))].">
      <j1 rule="back_rewrite" parents="1615"/>
      <j2 rule="rewrite" parents="1737"/>
    </justification>
  </clause>

  <clause id="1786">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_rewrite(1446),rewrite(1737(5),1737(6),785(3),1737(6))].">
      <j1 rule="back_rewrite" parents="1446"/>
      <j2 rule="rewrite" parents="1737 1737 785 1737"/>
    </justification>
  </clause>

  <clause id="1792">
    <literal><![CDATA[
      (((x ^ (y v x)) v z) v x) ^ (((((x ^ (y v x)) v z) v u) v v) v w) = (x ^ (y v x)) v z
    ]]></literal>
    <justification jstring="[back_rewrite(1780),rewrite(1786(1),1786(1),1786(4),1786(5),1786(5),1786(8),1786(8),1786(11),1786(12),1786(12),1786(15),1786(15),1786(18))].">
      <j1 rule="back_rewrite" parents="1780"/>
      <j2 rule="rewrite" parents="1786 1786 1786 1786 1786 1786 1786 1786 1786 1786 1786 1786 1786"/>
    </justification>
  </clause>

  <clause id="1814">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_rewrite(1737),rewrite(1786(5))].">
      <j1 rule="back_rewrite" parents="1737"/>
      <j2 rule="rewrite" parents="1786"/>
    </justification>
  </clause>

  <clause id="1831">
    <literal><![CDATA[
      (x ^ y) ^ (y v z) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(1700),rewrite(1786(3),1786(2),1786(6))].">
      <j1 rule="back_rewrite" parents="1700"/>
      <j2 rule="rewrite" parents="1786 1786 1786"/>
    </justification>
  </clause>

  <clause id="1832">
    <literal><![CDATA[
      x ^ (((y v x) ^ (z v x)) v u) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1699),rewrite(1786(1),1786(1),1786(2),1814(2),1786(2),1786(3),1814(3),1786(6))].">
      <j1 rule="back_rewrite" parents="1699"/>
      <j2 rule="rewrite" parents="1786 1786 1786 1814 1786 1786 1814 1786"/>
    </justification>
  </clause>

  <clause id="1834">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1696),rewrite(1786(1),1786(3),1786(4))].">
      <j1 rule="back_rewrite" parents="1696"/>
      <j2 rule="rewrite" parents="1786 1786 1786"/>
    </justification>
  </clause>

  <clause id="1836">
    <literal><![CDATA[
      (x ^ y) ^ (x v z) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(1694),rewrite(1786(1),1786(1),1786(3),1786(3),1814(4),1786(2),1786(2),1786(3),1786(2),1786(4),1786(4))].">
      <j1 rule="back_rewrite" parents="1694"/>
      <j2 rule="rewrite" parents="1786 1786 1786 1786 1814 1786 1786 1786 1786 1786 1786"/>
    </justification>
  </clause>

  <clause id="1848">
    <literal><![CDATA[
      (x v (((x ^ y) v x) ^ z)) ^ (((u v x) ^ (v v x)) v w) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1681),rewrite(1786(1),1786(2),1814(2),1786(1),1786(5),1786(6),1786(10))].">
      <j1 rule="back_rewrite" parents="1681"/>
      <j2 rule="rewrite" parents="1786 1786 1814 1786 1786 1786 1786"/>
    </justification>
  </clause>

  <clause id="1861">
    <literal><![CDATA[
      (x v (((x ^ y) v (z ^ x)) ^ u)) ^ (x v v) = x
    ]]></literal>
    <justification jstring="[back_rewrite(1664),rewrite(1786(1),1786(2),1814(2),1786(1),1786(2),1786(6),1786(7),1814(7),1786(8))].">
      <j1 rule="back_rewrite" parents="1664"/>
      <j2 rule="rewrite" parents="1786 1786 1814 1786 1786 1786 1786 1814 1786"/>
    </justification>
  </clause>

  <clause id="2195">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (x v z) = x
    ]]></literal>
    <justification jstring="[back_rewrite(682),rewrite(1786(5))].">
      <j1 rule="back_rewrite" parents="682"/>
      <j2 rule="rewrite" parents="1786"/>
    </justification>
  </clause>

  <clause id="2299">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_rewrite(1784),rewrite(1814(2),1814(3),1814(2),1786(1),1814(3))].">
      <j1 rule="back_rewrite" parents="1784"/>
      <j2 rule="rewrite" parents="1814 1814 1814 1786 1814"/>
    </justification>
  </clause>

  <clause id="2328">
    <literal><![CDATA[
      ((x ^ (y v x)) v z) v x = (x ^ (y v x)) v z
    ]]></literal>
    <justification jstring="[para(1792(a,1),1834(a,1)),flip(a)].">
      <j1 rule="para" parents="1792 1834"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2330">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(1786(a,1),1836(a,1,2))].">
      <j1 rule="para" parents="1786 1836"/>
    </justification>
  </clause>

  <clause id="2332">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(1814(a,1),2330(a,1,1)),rewrite(1814(4))].">
      <j1 rule="para" parents="1814 2330"/>
      <j2 rule="rewrite" parents="1814"/>
    </justification>
  </clause>

  <clause id="2334">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(2328),rewrite(2332(2),2332(4))].">
      <j1 rule="back_rewrite" parents="2328"/>
      <j2 rule="rewrite" parents="2332 2332"/>
    </justification>
  </clause>

  <clause id="2360">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(2195(a,1),2299(a,1)),flip(a)].">
      <j1 rule="para" parents="2195 2299"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2427">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(2360(a,1),1814(a,1,1))].">
      <j1 rule="para" parents="2360 1814"/>
    </justification>
  </clause>

  <clause id="2464">
    <literal><![CDATA[
      (x v y) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(1831(a,1),2427(a,1,2)),rewrite(1831(6))].">
      <j1 rule="para" parents="1831 2427"/>
      <j2 rule="rewrite" parents="1831"/>
    </justification>
  </clause>

  <clause id="2522">
    <literal><![CDATA[
      x ^ ((y v x) ^ (z v x)) = x
    ]]></literal>
    <justification jstring="[para(1786(a,1),1832(a,1,2))].">
      <j1 rule="para" parents="1786 1832"/>
    </justification>
  </clause>

  <clause id="2541">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[para(2334(a,1),2522(a,1,2,2))].">
      <j1 rule="para" parents="2334 2522"/>
    </justification>
  </clause>

  <clause id="2573">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(2541(a,1),2427(a,1,2)),rewrite(2541(8))].">
      <j1 rule="para" parents="2541 2427"/>
      <j2 rule="rewrite" parents="2541"/>
    </justification>
  </clause>

  <clause id="2578">
    <literal><![CDATA[
      c1 v (c2 ^ (c1 ^ c3)) != c1
    ]]></literal>
    <literal><![CDATA[
      c1 ^ (c2 v (c1 v c3)) != c1
    ]]></literal>
    <literal><![CDATA[
      ((c2 ^ c1) v (c1 ^ c3)) v c1 != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[back_rewrite(3),rewrite(2573(38)),xx(d)].">
      <j1 rule="back_rewrite" parents="3"/>
      <j2 rule="rewrite" parents="2573"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="2834">
    <literal><![CDATA[
      x v (((x ^ y) v x) ^ z) = x
    ]]></literal>
    <justification jstring="[para(1848(a,1),2332(a,1)),flip(a)].">
      <j1 rule="para" parents="1848 2332"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2859">
    <literal><![CDATA[
      x v ((x ^ y) v x) = x
    ]]></literal>
    <justification jstring="[para(1749(a,1),2834(a,1,2))].">
      <j1 rule="para" parents="1749 2834"/>
    </justification>
  </clause>

  <clause id="2867">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2464(a,1),2834(a,1,2))].">
      <j1 rule="para" parents="2464 2834"/>
    </justification>
  </clause>

  <clause id="2903">
    <literal><![CDATA[
      c1 ^ (c2 v (c1 v c3)) != c1
    ]]></literal>
    <literal><![CDATA[
      ((c2 ^ c1) v (c1 ^ c3)) v c1 != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[back_rewrite(2578),rewrite(2867(7)),xx(a)].">
      <j1 rule="back_rewrite" parents="2578"/>
      <j2 rule="rewrite" parents="2867"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="2904">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(2859(a,1),1814(a,1,1)),rewrite(2332(3)),flip(a)].">
      <j1 rule="para" parents="2859 1814"/>
      <j2 rule="rewrite" parents="2332"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2925">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1814(a,1),2904(a,1,1))].">
      <j1 rule="para" parents="1814 2904"/>
    </justification>
  </clause>

  <clause id="3048">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(2925(a,1),1834(a,1,2))].">
      <j1 rule="para" parents="2925 1834"/>
    </justification>
  </clause>

  <clause id="3095">
    <literal><![CDATA[
      ((c2 ^ c1) v (c1 ^ c3)) v c1 != c1
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[back_rewrite(2903),rewrite(3048(7)),xx(a)].">
      <j1 rule="back_rewrite" parents="2903"/>
      <j2 rule="rewrite" parents="3048"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="3169">
    <literal><![CDATA[
      x v (((x ^ y) v (z ^ x)) ^ u) = x
    ]]></literal>
    <justification jstring="[para(1861(a,1),2299(a,1)),flip(a)].">
      <j1 rule="para" parents="1861 2299"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7861">
    <literal><![CDATA[
      x v ((x ^ y) v (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(1749(a,1),3169(a,1,2))].">
      <j1 rule="para" parents="1749 3169"/>
    </justification>
  </clause>

  <clause id="8038">
    <literal><![CDATA[
      x v ((x ^ y) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2330(a,1),7861(a,1,2,2))].">
      <j1 rule="para" parents="2330 7861"/>
    </justification>
  </clause>

  <clause id="8354">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2427(a,1),8038(a,1,2,1))].">
      <j1 rule="para" parents="2427 8038"/>
    </justification>
  </clause>

  <clause id="8768">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[para(8354(a,1),2925(a,1,2)),rewrite(8354(8))].">
      <j1 rule="para" parents="8354 2925"/>
      <j2 rule="rewrite" parents="8354"/>
    </justification>
  </clause>

  <clause id="8769">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[resolve(8768,a,3095,a)].">
      <j1 rule="resolve" parents="8768 3095"/>
    </justification>
  </clause>

</proof>

</proofs>
