<?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 June-2006C, June 2006.
Process 13093 was started by mccune on cleo.thornwood,
Mon Jun 19 16:41:16 2006
The command was "/home/mccune/bin/prover9 -f a2.in".
]]></heading>

<proof number="1" length="100" max_count="89">

<comments><![CDATA[
% Proof 1 at 4.31 (+ 0.02) seconds: McKenzie.
% Length of proof is 100.
% Level of proof is 32.
% Maximum clause weight is 89.
% Given clauses 350.
]]></comments>

  <clause id="3">
    <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="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="4">
    <literal><![CDATA[
      B v (A ^ (B ^ C)) != B
    ]]></literal>
    <literal><![CDATA[
      B ^ (A v (B v C)) != B
    ]]></literal>
    <literal><![CDATA[
      ((A ^ B) v (B ^ C)) v B != B
    ]]></literal>
    <literal><![CDATA[
      ((A v B) ^ (B v C)) ^ B != B
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="5">
    <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(3(a,1),3(a,1,1,2,1,1))].">
      <j1 rule="para" parents="3 3"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (((z v y) ^ (u v y)) v v) = y
    ]]></literal>
    <justification jstring="[para(3(a,1),3(a,1,1,2))].">
      <j1 rule="para" parents="3 3"/>
    </justification>
  </clause>

  <clause id="8">
    <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(3(a,1),3(a,1,2,1))].">
      <j1 rule="para" parents="3 3"/>
    </justification>
  </clause>

  <clause id="11">
    <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(7(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="7 7"/>
    </justification>
  </clause>

  <clause id="12">
    <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(11(a,1),3(a,1,1,2,1,1))].">
      <j1 rule="para" parents="11 3"/>
    </justification>
  </clause>

  <clause id="18">
    <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(7(a,1),12(a,1,2,1))].">
      <j1 rule="para" parents="7 12"/>
    </justification>
  </clause>

  <clause id="22">
    <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(3(a,1),5(a,1,1,2))].">
      <j1 rule="para" parents="3 5"/>
    </justification>
  </clause>

  <clause id="26">
    <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(12(a,1),5(a,1,1,1,1,2)),demod(12(17),12(22),12(26),12(27),12(31))].">
      <j1 rule="para" parents="12 5"/>
      <j2 rule="demod" parents="12 12 12 12 12"/>
    </justification>
  </clause>

  <clause id="30">
    <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(11(a,1),18(a,1,1,2,1,2))].">
      <j1 rule="para" parents="11 18"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (((u v y) ^ (v v y)) v w) = y
    ]]></literal>
    <justification jstring="[para(3(a,1),26(a,1,1,2))].">
      <j1 rule="para" parents="3 26"/>
    </justification>
  </clause>

  <clause id="42">
    <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(11(a,1),26(a,1,1,2,1,1,1,2))].">
      <j1 rule="para" parents="11 26"/>
    </justification>
  </clause>

  <clause id="51">
    <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(7(a,1),38(a,1,2,1))].">
      <j1 rule="para" parents="7 38"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      (((x v (y ^ z)) ^ (y ^ z)) v (u ^ (y ^ z))) ^ (z v v) = y ^ z
    ]]></literal>
    <justification jstring="[para(38(a,1),38(a,1,2,1))].">
      <j1 rule="para" parents="38 38"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (y v y)) ^ (y v z) = y v y
    ]]></literal>
    <justification jstring="[para(11(a,1),51(a,1,1,2))].">
      <j1 rule="para" parents="11 51"/>
    </justification>
  </clause>

  <clause id="74">
    <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(71(a,1),7(a,1,1,1))].">
      <j1 rule="para" parents="71 7"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      ((x v x) v (y ^ (x v x))) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(71(a,1),51(a,1,1,1))].">
      <j1 rule="para" parents="71 51"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      ((x v x) v (x v x)) ^ (x v y) = x v x
    ]]></literal>
    <justification jstring="[para(71(a,1),71(a,1,1,1))].">
      <j1 rule="para" parents="71 71"/>
    </justification>
  </clause>

  <clause id="97">
    <literal><![CDATA[
      ((x v x) v (((y ^ (x v x)) v (z ^ x)) ^ u)) ^ (x v v) = x v x
    ]]></literal>
    <justification jstring="[para(8(a,1),87(a,1,1,2))].">
      <j1 rule="para" parents="8 87"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ ((((u v y) ^ (v v y)) v w) v v6) = y
    ]]></literal>
    <justification jstring="[para(3(a,1),61(a,1,1,1,1,2)),demod(3(14),3(15),3(23))].">
      <j1 rule="para" parents="3 61"/>
      <j2 rule="demod" parents="3 3 3"/>
    </justification>
  </clause>

  <clause id="107">
    <literal><![CDATA[
      (((x v (y v y)) ^ (y v y)) v (z ^ y)) ^ (y v u) = y v y
    ]]></literal>
    <justification jstring="[para(61(a,1),51(a,1,1,2))].">
      <j1 rule="para" parents="61 51"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      ((x v x) v (y ^ x)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(61(a,1),87(a,1,1,2))].">
      <j1 rule="para" parents="61 87"/>
    </justification>
  </clause>

  <clause id="116">
    <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(11(a,1),108(a,1,1,2))].">
      <j1 rule="para" parents="11 108"/>
    </justification>
  </clause>

  <clause id="148">
    <literal><![CDATA[
      ((x v x) v (((y ^ x) v (z ^ x)) ^ u)) ^ (x v v) = x v x
    ]]></literal>
    <justification jstring="[para(61(a,1),97(a,1,1,2,1,1))].">
      <j1 rule="para" parents="61 97"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      ((x v x) v x) ^ (x v y) = x v x
    ]]></literal>
    <justification jstring="[para(3(a,1),148(a,1,1,2))].">
      <j1 rule="para" parents="3 148"/>
    </justification>
  </clause>

  <clause id="179">
    <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(165(a,1),3(a,1,2,1))].">
      <j1 rule="para" parents="165 3"/>
    </justification>
  </clause>

  <clause id="180">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ ((y v y) v z) = y
    ]]></literal>
    <justification jstring="[para(165(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="165 7"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (((y v y) v u) v v) = y
    ]]></literal>
    <justification jstring="[para(180(a,1),61(a,1,1,1,1,2)),demod(180(8),180(9),180(15))].">
      <j1 rule="para" parents="180 61"/>
      <j2 rule="demod" parents="180 180 180"/>
    </justification>
  </clause>

  <clause id="232">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ ((((y v y) v u) v v) v w) = y
    ]]></literal>
    <justification jstring="[para(202(a,1),61(a,1,1,1,1,2)),demod(202(9),202(10),202(17))].">
      <j1 rule="para" parents="202 61"/>
      <j2 rule="demod" parents="202 202 202"/>
    </justification>
  </clause>

  <clause id="600">
    <literal><![CDATA[
      (((x v y) ^ y) v (((z ^ y) v (u ^ y)) ^ v)) ^ ((y v y) v w) = y
    ]]></literal>
    <justification jstring="[para(61(a,1),179(a,1,1,2,1,1))].">
      <j1 rule="para" parents="61 179"/>
    </justification>
  </clause>

  <clause id="618">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ ((y v y) v z) = y
    ]]></literal>
    <justification jstring="[para(3(a,1),600(a,1,1,2))].">
      <j1 rule="para" parents="3 600"/>
    </justification>
  </clause>

  <clause id="646">
    <literal><![CDATA[
      (((x v y) ^ y) v (y v y)) ^ (y v z) = y
    ]]></literal>
    <justification jstring="[para(618(a,1),7(a,1,2,1))].">
      <j1 rule="para" parents="618 7"/>
    </justification>
  </clause>

  <clause id="651">
    <literal><![CDATA[
      (((x v y) ^ y) v (z ^ y)) ^ (y v u) = y
    ]]></literal>
    <justification jstring="[para(618(a,1),38(a,1,2,1))].">
      <j1 rule="para" parents="618 38"/>
    </justification>
  </clause>

  <clause id="664">
    <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(646(a,1),18(a,1,1,1))].">
      <j1 rule="para" parents="646 18"/>
    </justification>
  </clause>

  <clause id="666">
    <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(646(a,1),30(a,1,1,1))].">
      <j1 rule="para" parents="646 30"/>
    </justification>
  </clause>

  <clause id="676">
    <literal><![CDATA[
      (x v (x v x)) ^ (x v y) = x v x
    ]]></literal>
    <justification jstring="[para(646(a,1),71(a,1,1,1))].">
      <j1 rule="para" parents="646 71"/>
    </justification>
  </clause>

  <clause id="677">
    <literal><![CDATA[
      (x v x) v (x v x) = x v x
    ]]></literal>
    <justification jstring="[para(71(a,1),646(a,1,1,1)),demod(676(8))].">
      <j1 rule="para" parents="71 646"/>
      <j2 rule="demod" parents="676"/>
    </justification>
  </clause>

  <clause id="683">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(646(a,1),107(a,1,1,1))].">
      <j1 rule="para" parents="646 107"/>
    </justification>
  </clause>

  <clause id="696">
    <literal><![CDATA[
      (x v (y ^ (x v x))) ^ ((((x v x) v z) v u) v v) = x v x
    ]]></literal>
    <justification jstring="[para(646(a,1),232(a,1,1,1)),demod(677(6))].">
      <j1 rule="para" parents="646 232"/>
      <j2 rule="demod" parents="677"/>
    </justification>
  </clause>

  <clause id="734">
    <literal><![CDATA[
      (((x v y) ^ y) v ((y v (z ^ y)) ^ u)) ^ ((y v y) v v) = y
    ]]></literal>
    <justification jstring="[para(646(a,1),179(a,1,1,2,1,1))].">
      <j1 rule="para" parents="646 179"/>
    </justification>
  </clause>

  <clause id="740">
    <literal><![CDATA[
      (x v ((x v x) ^ y)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[back_demod(666),demod(677(3),677(3))].">
      <j1 rule="back_demod" parents="666"/>
      <j2 rule="demod" parents="677 677"/>
    </justification>
  </clause>

  <clause id="742">
    <literal><![CDATA[
      (x v (((x v x) v (y ^ (x v x))) ^ z)) ^ (x v u) = x v x
    ]]></literal>
    <justification jstring="[back_demod(664),demod(677(3))].">
      <j1 rule="back_demod" parents="664"/>
      <j2 rule="demod" parents="677"/>
    </justification>
  </clause>

  <clause id="786">
    <literal><![CDATA[
      (x v x) ^ (x v y) = x v x
    ]]></literal>
    <justification jstring="[back_demod(88),demod(677(3))].">
      <j1 rule="back_demod" parents="88"/>
      <j2 rule="demod" parents="677"/>
    </justification>
  </clause>

  <clause id="791">
    <literal><![CDATA[
      (x v x) ^ (((y v (x v x)) ^ (z v (x v x))) v u) = x v x
    ]]></literal>
    <justification jstring="[back_demod(74),demod(677(4),677(3))].">
      <j1 rule="back_demod" parents="74"/>
      <j2 rule="demod" parents="677 677"/>
    </justification>
  </clause>

  <clause id="883">
    <literal><![CDATA[
      ((x v x) v ((y ^ x) v (y ^ x))) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(786(a,1),148(a,1,1,2))].">
      <j1 rule="para" parents="786 148"/>
    </justification>
  </clause>

  <clause id="913">
    <literal><![CDATA[
      (((x v y) ^ y) v y) ^ (((z v y) ^ (u v y)) v v) = y
    ]]></literal>
    <justification jstring="[para(38(a,1),22(a,1,1,1,1,2)),demod(38(13),38(15),38(16),38(20))].">
      <j1 rule="para" parents="38 22"/>
      <j2 rule="demod" parents="38 38 38 38"/>
    </justification>
  </clause>

  <clause id="1012">
    <literal><![CDATA[
      ((x v x) v ((x v x) ^ y)) ^ ((x v x) v z) = x v x
    ]]></literal>
    <justification jstring="[para(677(a,1),740(a,1,1,2,1)),demod(677(10))].">
      <j1 rule="para" parents="677 740"/>
      <j2 rule="demod" parents="677"/>
    </justification>
  </clause>

  <clause id="1447">
    <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(38(a,1),1012(a,1,1,2))].">
      <j1 rule="para" parents="38 1012"/>
    </justification>
  </clause>

  <clause id="1514">
    <literal><![CDATA[
      (x v (y ^ x)) ^ ((((x v x) v z) v u) v v) = x v x
    ]]></literal>
    <justification jstring="[para(61(a,1),696(a,1,1,2))].">
      <j1 rule="para" parents="61 696"/>
    </justification>
  </clause>

  <clause id="1599">
    <literal><![CDATA[
      (((x v y) ^ y) v ((y v (z ^ y)) ^ u)) ^ (y v y) = y
    ]]></literal>
    <justification jstring="[para(677(a,1),734(a,1,2))].">
      <j1 rule="para" parents="677 734"/>
    </justification>
  </clause>

  <clause id="1616">
    <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(3(a,1),1599(a,1,1,2))].">
      <j1 rule="para" parents="3 1599"/>
    </justification>
  </clause>

  <clause id="1642">
    <literal><![CDATA[
      (x v (((x v x) v x) ^ y)) ^ (x v z) = x v x
    ]]></literal>
    <justification jstring="[para(646(a,1),742(a,1,1,2,1,2))].">
      <j1 rule="para" parents="646 742"/>
    </justification>
  </clause>

  <clause id="1665">
    <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(677(a,1),42(a,1,2,1,2)),demod(677(7),677(7))].">
      <j1 rule="para" parents="677 42"/>
      <j2 rule="demod" parents="677 677"/>
    </justification>
  </clause>

  <clause id="1682">
    <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(1599(a,1),42(a,1,1,2,1,2)),demod(677(7),677(7))].">
      <j1 rule="para" parents="1599 42"/>
      <j2 rule="demod" parents="677 677"/>
    </justification>
  </clause>

  <clause id="1695">
    <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(1642(a,1),913(a,1,2,1))].">
      <j1 rule="para" parents="1642 913"/>
    </justification>
  </clause>

  <clause id="1697">
    <literal><![CDATA[
      (x v x) ^ (((x v y) v (x v y)) v z) = x v x
    ]]></literal>
    <justification jstring="[para(116(a,1),791(a,1,2,1))].">
      <j1 rule="para" parents="116 791"/>
    </justification>
  </clause>

  <clause id="1698">
    <literal><![CDATA[
      (x v x) ^ (((x v x) ^ (y v (x v x))) v z) = x v x
    ]]></literal>
    <justification jstring="[para(677(a,1),791(a,1,2,1,1))].">
      <j1 rule="para" parents="677 791"/>
    </justification>
  </clause>

  <clause id="1700">
    <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(22(a,1),791(a,1,2,1))].">
      <j1 rule="para" parents="22 791"/>
    </justification>
  </clause>

  <clause id="1701">
    <literal><![CDATA[
      ((x ^ y) v (x ^ y)) ^ ((y v y) v z) = (x ^ y) v (x ^ y)
    ]]></literal>
    <justification jstring="[para(883(a,1),791(a,1,2,1))].">
      <j1 rule="para" parents="883 791"/>
    </justification>
  </clause>

  <clause id="1738">
    <literal><![CDATA[
      ((x v y) ^ y) v ((x v y) ^ y) = y
    ]]></literal>
    <justification jstring="[para(38(a,1),1698(a,1,2,1)),demod(651(7)),flip(a)].">
      <j1 rule="para" parents="38 1698"/>
      <j2 rule="demod" parents="651"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1750">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[para(99(a,1),1698(a,1,2,1)),demod(1738(5),1738(7))].">
      <j1 rule="para" parents="99 1698"/>
      <j2 rule="demod" parents="1738 1738"/>
    </justification>
  </clause>

  <clause id="1781">
    <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(1698(a,1),1514(a,1,1,2))].">
      <j1 rule="para" parents="1698 1514"/>
    </justification>
  </clause>

  <clause id="1785">
    <literal><![CDATA[
      (((x v ((y v z) ^ z)) ^ ((y v z) ^ z)) v z) ^ z = (y v z) ^ z
    ]]></literal>
    <justification jstring="[back_demod(1616),demod(1738(12))].">
      <j1 rule="back_demod" parents="1616"/>
      <j2 rule="demod" parents="1738"/>
    </justification>
  </clause>

  <clause id="1787">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[back_demod(1447),demod(1738(5),1738(6),786(3),1738(6))].">
      <j1 rule="back_demod" parents="1447"/>
      <j2 rule="demod" parents="1738 1738 786 1738"/>
    </justification>
  </clause>

  <clause id="1793">
    <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_demod(1781),demod(1787(1),1787(1),1787(4),1787(5),1787(5),1787(8),1787(8),1787(11),1787(12),1787(12),1787(15),1787(15),1787(18))].">
      <j1 rule="back_demod" parents="1781"/>
      <j2 rule="demod" parents="1787 1787 1787 1787 1787 1787 1787 1787 1787 1787 1787 1787 1787"/>
    </justification>
  </clause>

  <clause id="1815">
    <literal><![CDATA[
      (x v y) ^ y = y
    ]]></literal>
    <justification jstring="[back_demod(1738),demod(1787(5))].">
      <j1 rule="back_demod" parents="1738"/>
      <j2 rule="demod" parents="1787"/>
    </justification>
  </clause>

  <clause id="1832">
    <literal><![CDATA[
      (x ^ y) ^ (y v z) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(1701),demod(1787(3),1787(2),1787(6))].">
      <j1 rule="back_demod" parents="1701"/>
      <j2 rule="demod" parents="1787 1787 1787"/>
    </justification>
  </clause>

  <clause id="1833">
    <literal><![CDATA[
      x ^ (((y v x) ^ (z v x)) v u) = x
    ]]></literal>
    <justification jstring="[back_demod(1700),demod(1787(1),1787(1),1787(2),1815(2),1787(2),1787(3),1815(3),1787(6))].">
      <j1 rule="back_demod" parents="1700"/>
      <j2 rule="demod" parents="1787 1787 1787 1815 1787 1787 1815 1787"/>
    </justification>
  </clause>

  <clause id="1835">
    <literal><![CDATA[
      x ^ ((x v y) v z) = x
    ]]></literal>
    <justification jstring="[back_demod(1697),demod(1787(1),1787(3),1787(4))].">
      <j1 rule="back_demod" parents="1697"/>
      <j2 rule="demod" parents="1787 1787 1787"/>
    </justification>
  </clause>

  <clause id="1837">
    <literal><![CDATA[
      (x ^ y) ^ (x v z) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(1695),demod(1787(1),1787(1),1787(3),1787(3),1815(4),1787(2),1787(2),1787(3),1787(2),1787(4),1787(4))].">
      <j1 rule="back_demod" parents="1695"/>
      <j2 rule="demod" parents="1787 1787 1787 1787 1815 1787 1787 1787 1787 1787 1787"/>
    </justification>
  </clause>

  <clause id="1849">
    <literal><![CDATA[
      (x v (((x ^ y) v x) ^ z)) ^ (((u v x) ^ (v v x)) v w) = x
    ]]></literal>
    <justification jstring="[back_demod(1682),demod(1787(1),1787(2),1815(2),1787(1),1787(5),1787(6),1787(10))].">
      <j1 rule="back_demod" parents="1682"/>
      <j2 rule="demod" parents="1787 1787 1815 1787 1787 1787 1787"/>
    </justification>
  </clause>

  <clause id="1862">
    <literal><![CDATA[
      (x v (((x ^ y) v (z ^ x)) ^ u)) ^ (x v v) = x
    ]]></literal>
    <justification jstring="[back_demod(1665),demod(1787(1),1787(2),1815(2),1787(1),1787(2),1787(6),1787(7),1815(7),1787(8))].">
      <j1 rule="back_demod" parents="1665"/>
      <j2 rule="demod" parents="1787 1787 1815 1787 1787 1787 1787 1815 1787"/>
    </justification>
  </clause>

  <clause id="2196">
    <literal><![CDATA[
      (x v (y ^ x)) ^ (x v z) = x
    ]]></literal>
    <justification jstring="[back_demod(683),demod(1787(5))].">
      <j1 rule="back_demod" parents="683"/>
      <j2 rule="demod" parents="1787"/>
    </justification>
  </clause>

  <clause id="2300">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[back_demod(1785),demod(1815(2),1815(3),1815(2),1787(1),1815(3))].">
      <j1 rule="back_demod" parents="1785"/>
      <j2 rule="demod" parents="1815 1815 1815 1787 1815"/>
    </justification>
  </clause>

  <clause id="2329">
    <literal><![CDATA[
      ((x ^ (y v x)) v z) v x = (x ^ (y v x)) v z
    ]]></literal>
    <justification jstring="[para(1793(a,1),1835(a,1)),flip(a)].">
      <j1 rule="para" parents="1793 1835"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2331">
    <literal><![CDATA[
      (x ^ y) ^ x = x ^ y
    ]]></literal>
    <justification jstring="[para(1787(a,1),1837(a,1,2))].">
      <j1 rule="para" parents="1787 1837"/>
    </justification>
  </clause>

  <clause id="2333">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(1815(a,1),2331(a,1,1)),demod(1815(4))].">
      <j1 rule="para" parents="1815 2331"/>
      <j2 rule="demod" parents="1815"/>
    </justification>
  </clause>

  <clause id="2335">
    <literal><![CDATA[
      (x v y) v x = x v y
    ]]></literal>
    <justification jstring="[back_demod(2329),demod(2333(2),2333(4))].">
      <j1 rule="back_demod" parents="2329"/>
      <j2 rule="demod" parents="2333 2333"/>
    </justification>
  </clause>

  <clause id="2361">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(2196(a,1),2300(a,1)),flip(a)].">
      <j1 rule="para" parents="2196 2300"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2428">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(2361(a,1),1815(a,1,1))].">
      <j1 rule="para" parents="2361 1815"/>
    </justification>
  </clause>

  <clause id="2465">
    <literal><![CDATA[
      (x v y) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(1832(a,1),2428(a,1,2)),demod(1832(6))].">
      <j1 rule="para" parents="1832 2428"/>
      <j2 rule="demod" parents="1832"/>
    </justification>
  </clause>

  <clause id="2523">
    <literal><![CDATA[
      x ^ ((y v x) ^ (z v x)) = x
    ]]></literal>
    <justification jstring="[para(1787(a,1),1833(a,1,2))].">
      <j1 rule="para" parents="1787 1833"/>
    </justification>
  </clause>

  <clause id="2542">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z)) = x
    ]]></literal>
    <justification jstring="[para(2335(a,1),2523(a,1,2,2))].">
      <j1 rule="para" parents="2335 2523"/>
    </justification>
  </clause>

  <clause id="2574">
    <literal><![CDATA[
      ((x v y) ^ (y v z)) ^ y = y
    ]]></literal>
    <justification jstring="[para(2542(a,1),2428(a,1,2)),demod(2542(8))].">
      <j1 rule="para" parents="2542 2428"/>
      <j2 rule="demod" parents="2542"/>
    </justification>
  </clause>

  <clause id="2579">
    <literal><![CDATA[
      B v (A ^ (B ^ C)) != B
    ]]></literal>
    <literal><![CDATA[
      B ^ (A v (B v C)) != B
    ]]></literal>
    <literal><![CDATA[
      ((A ^ B) v (B ^ C)) v B != B
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[back_demod(4),demod(2574(38)),xx(d)].">
      <j1 rule="back_demod" parents="4"/>
      <j2 rule="demod" parents="2574"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="2835">
    <literal><![CDATA[
      x v (((x ^ y) v x) ^ z) = x
    ]]></literal>
    <justification jstring="[para(1849(a,1),2333(a,1)),flip(a)].">
      <j1 rule="para" parents="1849 2333"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2860">
    <literal><![CDATA[
      x v ((x ^ y) v x) = x
    ]]></literal>
    <justification jstring="[para(1750(a,1),2835(a,1,2))].">
      <j1 rule="para" parents="1750 2835"/>
    </justification>
  </clause>

  <clause id="2868">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2465(a,1),2835(a,1,2))].">
      <j1 rule="para" parents="2465 2835"/>
    </justification>
  </clause>

  <clause id="2904">
    <literal><![CDATA[
      B ^ (A v (B v C)) != B
    ]]></literal>
    <literal><![CDATA[
      ((A ^ B) v (B ^ C)) v B != B
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[back_demod(2579),demod(2868(7)),xx(a)].">
      <j1 rule="back_demod" parents="2579"/>
      <j2 rule="demod" parents="2868"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="2905">
    <literal><![CDATA[
      (x ^ y) v x = x
    ]]></literal>
    <justification jstring="[para(2860(a,1),1815(a,1,1)),demod(2333(3)),flip(a)].">
      <j1 rule="para" parents="2860 1815"/>
      <j2 rule="demod" parents="2333"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2926">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(1815(a,1),2905(a,1,1))].">
      <j1 rule="para" parents="1815 2905"/>
    </justification>
  </clause>

  <clause id="3049">
    <literal><![CDATA[
      x ^ (y v (x v z)) = x
    ]]></literal>
    <justification jstring="[para(2926(a,1),1835(a,1,2))].">
      <j1 rule="para" parents="2926 1835"/>
    </justification>
  </clause>

  <clause id="3096">
    <literal><![CDATA[
      ((A ^ B) v (B ^ C)) v B != B
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[back_demod(2904),demod(3049(7)),xx(a)].">
      <j1 rule="back_demod" parents="2904"/>
      <j2 rule="demod" parents="3049"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="3170">
    <literal><![CDATA[
      x v (((x ^ y) v (z ^ x)) ^ u) = x
    ]]></literal>
    <justification jstring="[para(1862(a,1),2300(a,1)),flip(a)].">
      <j1 rule="para" parents="1862 2300"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7862">
    <literal><![CDATA[
      x v ((x ^ y) v (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(1750(a,1),3170(a,1,2))].">
      <j1 rule="para" parents="1750 3170"/>
    </justification>
  </clause>

  <clause id="8039">
    <literal><![CDATA[
      x v ((x ^ y) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2331(a,1),7862(a,1,2,2))].">
      <j1 rule="para" parents="2331 7862"/>
    </justification>
  </clause>

  <clause id="8320">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(2428(a,1),8039(a,1,2,1))].">
      <j1 rule="para" parents="2428 8039"/>
    </justification>
  </clause>

  <clause id="8733">
    <literal><![CDATA[
      ((x ^ y) v (y ^ z)) v y = y
    ]]></literal>
    <justification jstring="[para(8320(a,1),2926(a,1,2)),demod(8320(8))].">
      <j1 rule="para" parents="8320 2926"/>
      <j2 rule="demod" parents="8320"/>
    </justification>
  </clause>

  <clause id="8734">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(McKenzie)
    ]]></attribute>
    <justification jstring="[resolve(8733,a,3096,a)].">
      <j1 rule="resolve" parents="8733 3096"/>
    </justification>
  </clause>

</proof>

</proofs>
