<?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>H27d.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13657 was started by mccune on cleo.thornwood,
Mon Jun 19 16:45:34 2006
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H27d.in".
]]></heading>

<proof number="1" length="96" max_count="25">

<comments><![CDATA[
% Proof 1 at 13.67 (+ 0.20) seconds.
% Length of proof is 96.
% Level of proof is 24.
% Maximum clause weight is 23.
% Given clauses 695.
]]></comments>

  <clause id="10">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v u))) = x ^ (y v ((x ^ z) v (z ^ (y v u))))
    ]]></literal>
    <attribute><![CDATA[
      label(H49)
    ]]></attribute>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

  <clause id="14">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

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

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

  <clause id="18">
    <literal><![CDATA[
      x v x ' = 1
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      x ^ x ' = 0
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x ' = y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x ^ (y v ((x ^ z) v (z ^ (y v u)))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <attribute><![CDATA[
      label(H49)
    ]]></attribute>
    <justification jstring="[copy(10),flip(a)].">
      <j1 rule="copy" parents="10"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      (c1 ^ c2) v (c1 ^ c3) != c1 ^ (c2 v c3)
    ]]></literal>
    <attribute><![CDATA[
      label(distributivity)
    ]]></attribute>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

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

  <clause id="24">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(14(a,1),15(a,1,1)),demod(15(2))].">
      <j1 rule="para" parents="14 15"/>
      <j2 rule="demod" parents="15"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      x ^ (y v x) = x
    ]]></literal>
    <justification jstring="[para(12(a,1),16(a,1,2))].">
      <j1 rule="para" parents="12 16"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(16(a,1),15(a,1,1)),flip(a)].">
      <j1 rule="para" parents="16 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x ^ (y ^ ((x ^ y) v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(16(a,1),15(a,1)),flip(a)].">
      <j1 rule="para" parents="16 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x v ((x ^ y) v z) = x v z
    ]]></literal>
    <justification jstring="[para(17(a,1),13(a,1,1)),flip(a)].">
      <j1 rule="para" parents="17 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(14(a,1),17(a,1,2))].">
      <j1 rule="para" parents="14 17"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(16(a,1),17(a,1,2))].">
      <j1 rule="para" parents="16 17"/>
    </justification>
  </clause>

  <clause id="35">
    <literal><![CDATA[
      x v (x ' v y) = 1 v y
    ]]></literal>
    <justification jstring="[para(18(a,1),13(a,1,1)),flip(a)].">
      <j1 rule="para" parents="18 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x v (y v (x v y) ') = 1
    ]]></literal>
    <justification jstring="[para(18(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="18 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x ^ 1 = x
    ]]></literal>
    <justification jstring="[para(18(a,1),16(a,1,2))].">
      <j1 rule="para" parents="18 16"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x ^ (x ' ^ y) = 0 ^ y
    ]]></literal>
    <justification jstring="[para(19(a,1),15(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      x ^ (y ^ (x ^ y) ') = 0
    ]]></literal>
    <justification jstring="[para(19(a,1),15(a,1)),flip(a)].">
      <j1 rule="para" parents="19 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x v 0 = x
    ]]></literal>
    <justification jstring="[para(19(a,1),17(a,1,2))].">
      <j1 rule="para" parents="19 17"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ x != 0
    ]]></literal>
    <literal><![CDATA[
      y ' = x
    ]]></literal>
    <justification jstring="[para(12(a,1),20(a,1))].">
      <j1 rule="para" parents="12 20"/>
    </justification>
  </clause>

  <clause id="44">
    <literal><![CDATA[
      (x ^ y) v z != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ (y ^ z) != 0
    ]]></literal>
    <literal><![CDATA[
      (x ^ y) ' = z
    ]]></literal>
    <justification jstring="[para(15(a,1),20(b,1))].">
      <j1 rule="para" parents="15 20"/>
    </justification>
  </clause>

  <clause id="49">
    <literal><![CDATA[
      x ^ (y v ((z ^ x) v (z ^ (y v u)))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(14(a,1),21(a,1,2,2,1))].">
      <j1 rule="para" parents="14 21"/>
    </justification>
  </clause>

  <clause id="53">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v y '))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(18(a,1),21(a,1,2,2,2,2)),demod(37(3),12(2),31(2)),flip(a)].">
      <j1 rule="para" parents="18 21"/>
      <j2 rule="demod" parents="37 12 31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="61">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(34(a,1),13(a,1,1)),flip(a)].">
      <j1 rule="para" parents="34 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      1 ^ x = x
    ]]></literal>
    <justification jstring="[para(37(a,1),14(a,1)),flip(a)].">
      <j1 rule="para" parents="37 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="71">
    <literal><![CDATA[
      0 v x = x
    ]]></literal>
    <justification jstring="[para(40(a,1),12(a,1)),flip(a)].">
      <j1 rule="para" parents="40 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x v (y v x ') = y v 1
    ]]></literal>
    <justification jstring="[para(18(a,1),23(a,1,2)),flip(a)].">
      <j1 rule="para" parents="18 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(68(a,1),16(a,1))].">
      <j1 rule="para" parents="68 16"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x v (x ' v y) = 1
    ]]></literal>
    <justification jstring="[back_demod(35),demod(84(5))].">
      <j1 rule="back_demod" parents="35"/>
      <j2 rule="demod" parents="84"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(71(a,1),16(a,1,2))].">
      <j1 rule="para" parents="71 16"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x ^ (x ' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_demod(38),demod(87(5))].">
      <j1 rule="back_demod" parents="38"/>
      <j2 rule="demod" parents="87"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(16(a,1),24(a,1,2)),flip(a)].">
      <j1 rule="para" parents="16 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      x ^ (y ^ x ') = y ^ 0
    ]]></literal>
    <justification jstring="[para(19(a,1),24(a,1,2)),flip(a)].">
      <j1 rule="para" parents="19 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="95">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(84(a,1),12(a,1)),flip(a)].">
      <j1 rule="para" parents="84 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x v (y v x ') = 1
    ]]></literal>
    <justification jstring="[back_demod(79),demod(95(5))].">
      <j1 rule="back_demod" parents="79"/>
      <j2 rule="demod" parents="95"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(87(a,1),14(a,1)),flip(a)].">
      <j1 rule="para" parents="87 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="99">
    <literal><![CDATA[
      x ^ (y ^ x ') = 0
    ]]></literal>
    <justification jstring="[back_demod(92),demod(98(5))].">
      <j1 rule="back_demod" parents="92"/>
      <j2 rule="demod" parents="98"/>
    </justification>
  </clause>

  <clause id="102">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(25(a,1),15(a,1,1)),flip(a)].">
      <j1 rule="para" parents="25 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="108">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(31(a,1),13(a,1,1)),flip(a)].">
      <j1 rule="para" parents="31 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="117">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v (y ' v u)))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(85(a,1),21(a,1,2,2,2,2)),demod(37(3),12(2),31(2)),flip(a)].">
      <j1 rule="para" parents="85 21"/>
      <j2 rule="demod" parents="37 12 31"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="149">
    <literal><![CDATA[
      x ^ (x v y) ' = 0
    ]]></literal>
    <justification jstring="[para(19(a,1),27(a,1,2)),demod(98(2)),flip(a)].">
      <j1 rule="para" parents="19 27"/>
      <j2 rule="demod" parents="98"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="154">
    <literal><![CDATA[
      x ^ ((x v y) ' ^ z) = 0
    ]]></literal>
    <justification jstring="[para(89(a,1),27(a,1,2)),demod(98(2)),flip(a)].">
      <j1 rule="para" parents="89 27"/>
      <j2 rule="demod" parents="98"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="156">
    <literal><![CDATA[
      x ^ (y v x) ' = 0
    ]]></literal>
    <justification jstring="[para(12(a,1),149(a,1,2,1))].">
      <j1 rule="para" parents="12 149"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x ^ (y v ((x v z) ' ^ (y v u))) = x ^ (y v ((x v z) ' ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(149(a,1),21(a,1,2,2,1)),demod(71(6))].">
      <j1 rule="para" parents="149 21"/>
      <j2 rule="demod" parents="71"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y)) ') = 0
    ]]></literal>
    <justification jstring="[para(156(a,1),15(a,1)),flip(a)].">
      <j1 rule="para" parents="156 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      x v (y ^ (x ^ y) ') != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ (x ^ y) ' = x '
    ]]></literal>
    <justification jstring="[para(39(a,1),20(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="39 20"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="197">
    <literal><![CDATA[
      x v (x ^ y) ' = 1
    ]]></literal>
    <justification jstring="[para(18(a,1),29(a,1,2)),demod(95(2)),flip(a)].">
      <j1 rule="para" parents="18 29"/>
      <j2 rule="demod" parents="95"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="205">
    <literal><![CDATA[
      x v (y v ((x ^ z) v y) ') = 1
    ]]></literal>
    <justification jstring="[para(36(a,1),29(a,1,2)),demod(95(2)),flip(a)].">
      <j1 rule="para" parents="36 29"/>
      <j2 rule="demod" parents="95"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="209">
    <literal><![CDATA[
      x v (y ^ x) ' = 1
    ]]></literal>
    <justification jstring="[para(14(a,1),197(a,1,2,1))].">
      <j1 rule="para" parents="14 197"/>
    </justification>
  </clause>

  <clause id="215">
    <literal><![CDATA[
      x v (y v (z ^ (x v y)) ') = 1
    ]]></literal>
    <justification jstring="[para(209(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="209 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      ((x v y) ^ z) v (x ^ z) ' = 1
    ]]></literal>
    <justification jstring="[para(27(a,1),209(a,1,2,1))].">
      <j1 rule="para" parents="27 209"/>
    </justification>
  </clause>

  <clause id="305">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      y ' = x
    ]]></literal>
    <justification jstring="[para(14(a,1),41(b,1))].">
      <j1 rule="para" parents="14 41"/>
    </justification>
  </clause>

  <clause id="308">
    <literal><![CDATA[
      x ' ' = x
    ]]></literal>
    <justification jstring="[para(18(a,1),41(a,1)),demod(14(5),19(5)),xx(a),xx(b)].">
      <j1 rule="para" parents="18 41"/>
      <j2 rule="demod" parents="14 19"/>
      <j2 rule="xx"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="327">
    <literal><![CDATA[
      x ' v (y v x) = 1
    ]]></literal>
    <justification jstring="[para(308(a,1),96(a,1,2,2))].">
      <j1 rule="para" parents="308 96"/>
    </justification>
  </clause>

  <clause id="328">
    <literal><![CDATA[
      x ' ^ (y ^ x) = 0
    ]]></literal>
    <justification jstring="[para(308(a,1),99(a,1,2,2))].">
      <j1 rule="para" parents="308 99"/>
    </justification>
  </clause>

  <clause id="333">
    <literal><![CDATA[
      x ' ^ (y v x) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(327(a,1),20(a,1)),demod(308(10)),flip(c),xx(a)].">
      <j1 rule="para" parents="327 20"/>
      <j2 rule="demod" parents="308"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="342">
    <literal><![CDATA[
      x ^ (y ^ (y ^ ((x ^ y) v z)) ') = 0
    ]]></literal>
    <justification jstring="[para(28(a,1),328(a,1,2)),demod(14(6),15(6))].">
      <j1 rule="para" parents="28 328"/>
      <j2 rule="demod" parents="14 15"/>
    </justification>
  </clause>

  <clause id="740">
    <literal><![CDATA[
      x ^ ((y ^ x) v (y ^ z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(71(a,1),49(a,1,2,2,2,2)),demod(71(5),71(8),90(7))].">
      <j1 rule="para" parents="71 49"/>
      <j2 rule="demod" parents="71 71 90"/>
    </justification>
  </clause>

  <clause id="826">
    <literal><![CDATA[
      x ' ^ (x v (y ^ x ')) = x ' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(34(a,1),53(a,1,2,2,2))].">
      <j1 rule="para" parents="34 53"/>
    </justification>
  </clause>

  <clause id="1372">
    <literal><![CDATA[
      x v ((x ^ y) v (x ^ z)) ' = 1
    ]]></literal>
    <justification jstring="[para(205(a,1),29(a,1)),flip(a)].">
      <j1 rule="para" parents="205 29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2824">
    <literal><![CDATA[
      x ' ^ (x v (y ^ (x ' v z))) = x ' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(61(a,1),117(a,1,2,2,2))].">
      <j1 rule="para" parents="61 117"/>
    </justification>
  </clause>

  <clause id="6985">
    <literal><![CDATA[
      x ' ^ (x v y) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(12(a,1),333(a,1,2))].">
      <j1 rule="para" parents="12 333"/>
    </justification>
  </clause>

  <clause id="10458">
    <literal><![CDATA[
      x ' ^ (x v (x ' v y) ') = 0
    ]]></literal>
    <justification jstring="[para(160(a,1),826(a,2)),demod(14(8),154(8),40(3),14(2),19(2),2824(10)),flip(a)].">
      <j1 rule="para" parents="160 826"/>
      <j2 rule="demod" parents="14 154 40 14 19 2824"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10472">
    <literal><![CDATA[
      x v (x ' v y) ' = x
    ]]></literal>
    <justification jstring="[hyper(6985,a,10458,a),demod(12(4))].">
      <j1 rule="hyper" parents="6985 10458"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="10482">
    <literal><![CDATA[
      x v (y v x ') ' = x
    ]]></literal>
    <justification jstring="[para(12(a,1),10472(a,1,2,1))].">
      <j1 rule="para" parents="12 10472"/>
    </justification>
  </clause>

  <clause id="10492">
    <literal><![CDATA[
      x ' v (x v y) ' = x '
    ]]></literal>
    <justification jstring="[para(308(a,1),10472(a,1,2,1,1))].">
      <j1 rule="para" parents="308 10472"/>
    </justification>
  </clause>

  <clause id="10619">
    <literal><![CDATA[
      x ' v (y v x) ' = x '
    ]]></literal>
    <justification jstring="[para(308(a,1),10482(a,1,2,1,2))].">
      <j1 rule="para" parents="308 10482"/>
    </justification>
  </clause>

  <clause id="10700">
    <literal><![CDATA[
      x ' ^ (x v y) ' = (x v y) '
    ]]></literal>
    <justification jstring="[para(10492(a,1),25(a,1,2)),demod(14(4))].">
      <j1 rule="para" parents="10492 25"/>
      <j2 rule="demod" parents="14"/>
    </justification>
  </clause>

  <clause id="10788">
    <literal><![CDATA[
      x ' v (x ^ y) ' = (x ^ y) '
    ]]></literal>
    <justification jstring="[para(17(a,1),10619(a,1,2,1)),demod(12(4))].">
      <j1 rule="para" parents="17 10619"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="11700">
    <literal><![CDATA[
      x ^ (y v (x ^ y ')) ' = 0
    ]]></literal>
    <justification jstring="[para(10700(a,1),164(a,1,2))].">
      <j1 rule="para" parents="10700 164"/>
    </justification>
  </clause>

  <clause id="11936">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' = x '
    ]]></literal>
    <justification jstring="[para(10788(a,1),16(a,1,2))].">
      <j1 rule="para" parents="10788 16"/>
    </justification>
  </clause>

  <clause id="11941">
    <literal><![CDATA[
      x v (x ' ^ y) ' = (x ' ^ y) '
    ]]></literal>
    <justification jstring="[para(308(a,1),10788(a,1,1))].">
      <j1 rule="para" parents="308 10788"/>
    </justification>
  </clause>

  <clause id="12019">
    <literal><![CDATA[
      x ' ^ (y ^ x) ' = x '
    ]]></literal>
    <justification jstring="[para(14(a,1),11936(a,1,2,1))].">
      <j1 rule="para" parents="14 11936"/>
    </justification>
  </clause>

  <clause id="12074">
    <literal><![CDATA[
      x v ((x ' v y) ^ (z ^ x) ') = 1
    ]]></literal>
    <justification jstring="[para(12019(a,1),222(a,1,2,1)),demod(308(7),12(6))].">
      <j1 rule="para" parents="12019 222"/>
      <j2 rule="demod" parents="308 12"/>
    </justification>
  </clause>

  <clause id="12929">
    <literal><![CDATA[
      (x ^ y) v (x ^ (x ^ y) ') = x
    ]]></literal>
    <justification jstring="[hyper(305,a,1372,a,b,11700,a),demod(308(7))].">
      <j1 rule="hyper" parents="305 1372 11700"/>
      <j2 rule="demod" parents="308"/>
    </justification>
  </clause>

  <clause id="12986">
    <literal><![CDATA[
      x v (y ' ^ (x v y)) ' = 1
    ]]></literal>
    <justification jstring="[para(11941(a,1),215(a,1,2))].">
      <j1 rule="para" parents="11941 215"/>
    </justification>
  </clause>

  <clause id="13653">
    <literal><![CDATA[
      (x ' ^ (x v y)) ' = (y ^ x ') '
    ]]></literal>
    <justification jstring="[hyper(44,a,12986,a,b,342,a),demod(12(7),826(8)),flip(a)].">
      <j1 rule="hyper" parents="44 12986 342"/>
      <j2 rule="demod" parents="12 826"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13737">
    <literal><![CDATA[
      ((x v y) ^ x ') ' = (y ^ x ') '
    ]]></literal>
    <justification jstring="[para(14(a,1),13653(a,1,1))].">
      <j1 rule="para" parents="14 13653"/>
    </justification>
  </clause>

  <clause id="13748">
    <literal><![CDATA[
      (x ^ (x ' v y)) ' = (y ^ x) '
    ]]></literal>
    <justification jstring="[para(308(a,1),13653(a,1,1,1)),demod(308(6))].">
      <j1 rule="para" parents="308 13653"/>
      <j2 rule="demod" parents="308"/>
    </justification>
  </clause>

  <clause id="13935">
    <literal><![CDATA[
      (x v y) ^ x ' = y ^ x '
    ]]></literal>
    <justification jstring="[para(13737(a,1),308(a,1,1)),demod(308(4)),flip(a)].">
      <j1 rule="para" parents="13737 308"/>
      <j2 rule="demod" parents="308"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14041">
    <literal><![CDATA[
      x ^ (x ' v y) = y ^ x
    ]]></literal>
    <justification jstring="[para(13748(a,1),308(a,1,1)),demod(308(3)),flip(a)].">
      <j1 rule="para" parents="13748 308"/>
      <j2 rule="demod" parents="308"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14084">
    <literal><![CDATA[
      (x ' v y) ^ (y ^ x) ' = x '
    ]]></literal>
    <justification jstring="[para(13748(a,1),171(a,1,2,2)),demod(12074(6),14041(8)),xx(a)].">
      <j1 rule="para" parents="13748 171"/>
      <j2 rule="demod" parents="12074 14041"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="14387">
    <literal><![CDATA[
      ((x ^ y) v z) ^ y ' = z ^ y '
    ]]></literal>
    <justification jstring="[para(108(a,1),13935(a,1,1)),demod(13935(3)),flip(a)].">
      <j1 rule="para" parents="108 13935"/>
      <j2 rule="demod" parents="13935"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14742">
    <literal><![CDATA[
      x ^ (x ^ y) ' = x ^ y '
    ]]></literal>
    <justification jstring="[para(14084(a,1),102(a,1,2)),flip(a)].">
      <j1 rule="para" parents="14084 102"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16629">
    <literal><![CDATA[
      (x ^ y) v (x ^ y ') = x
    ]]></literal>
    <justification jstring="[back_demod(12929),demod(14742(4))].">
      <j1 rule="back_demod" parents="12929"/>
      <j2 rule="demod" parents="14742"/>
    </justification>
  </clause>

  <clause id="20300">
    <literal><![CDATA[
      (x ^ y) v (y ^ x ') = y
    ]]></literal>
    <justification jstring="[para(14(a,1),16629(a,1,1))].">
      <j1 rule="para" parents="14 16629"/>
    </justification>
  </clause>

  <clause id="21106">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z ^ y ')) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(90(a,1),20300(a,1,1)),demod(15(5),13935(4))].">
      <j1 rule="para" parents="90 20300"/>
      <j2 rule="demod" parents="15 13935"/>
    </justification>
  </clause>

  <clause id="21134">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(740(a,1),20300(a,1,1)),demod(14387(6),15(4),21106(5)),flip(a)].">
      <j1 rule="para" parents="740 20300"/>
      <j2 rule="demod" parents="14387 15 21106"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21135">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(21134,a,22,a)].">
      <j1 rule="resolve" parents="21134 22"/>
    </justification>
  </clause>

</proof>

</proofs>
