<?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>H65d.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13659 was started by mccune on cleo.thornwood,
Mon Jun 19 16:45:47 2006
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H65d.in".
]]></heading>

<proof number="1" length="118" max_count="27">

<comments><![CDATA[
% Proof 1 at 166.57 (+ 1.98) seconds: distributivity.
% Length of proof is 118.
% Level of proof is 25.
% Maximum clause weight is 27.
% Given clauses 1707.
]]></comments>

  <clause id="10">
    <literal><![CDATA[
      x ^ (y v (z ^ u)) = x ^ (y v (x ^ ((x ^ y) v (z ^ u))))
    ]]></literal>
    <attribute><![CDATA[
      label(H65)
    ]]></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 ^ ((x ^ y) v (z ^ u)))) = x ^ (y v (z ^ u))
    ]]></literal>
    <attribute><![CDATA[
      label(H65)
    ]]></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[
      answer(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="26">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(13(a,1),16(a,1,2))].">
      <j1 rule="para" parents="13 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="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="32">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(15(a,1),17(a,1,2))].">
      <j1 rule="para" parents="15 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="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="42">
    <literal><![CDATA[
      x v (y v z) != 1
    ]]></literal>
    <literal><![CDATA[
      (x v y) ^ z != 0
    ]]></literal>
    <literal><![CDATA[
      (x v y) ' = z
    ]]></literal>
    <justification jstring="[para(13(a,1),20(a,1))].">
      <j1 rule="para" parents="13 20"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      x ^ (y v (x ^ ((x ^ y) v z))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(16(a,1),21(a,1,2,2,2,2)),demod(16(7))].">
      <j1 rule="para" parents="16 21"/>
      <j2 rule="demod" parents="16"/>
    </justification>
  </clause>

  <clause id="69">
    <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="75">
    <literal><![CDATA[
      x v (y v (x ^ z)) = y v x
    ]]></literal>
    <justification jstring="[para(17(a,1),23(a,1,2)),flip(a)].">
      <j1 rule="para" parents="17 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="76">
    <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="80">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(69(a,1),16(a,1))].">
      <j1 rule="para" parents="69 16"/>
    </justification>
  </clause>

  <clause id="81">
    <literal><![CDATA[
      x v (x ' v y) = 1
    ]]></literal>
    <justification jstring="[back_demod(35),demod(80(5))].">
      <j1 rule="back_demod" parents="35"/>
      <j2 rule="demod" parents="80"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      0 ' = 1
    ]]></literal>
    <justification jstring="[hyper(20,a,71,a,b,37,a)].">
      <j1 rule="hyper" parents="20 71 37"/>
    </justification>
  </clause>

  <clause id="83">
    <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="85">
    <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="86">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(24(a,1),17(a,1,2))].">
      <j1 rule="para" parents="24 17"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(80(a,1),12(a,1)),flip(a)].">
      <j1 rule="para" parents="80 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="90">
    <literal><![CDATA[
      x v (y v x ') = 1
    ]]></literal>
    <justification jstring="[back_demod(76),demod(89(5))].">
      <j1 rule="back_demod" parents="76"/>
      <j2 rule="demod" parents="89"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(83(a,1),14(a,1)),flip(a)].">
      <j1 rule="para" parents="83 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="96">
    <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="100">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = y ^ x
    ]]></literal>
    <justification jstring="[para(25(a,1),24(a,1,2)),flip(a)].">
      <j1 rule="para" parents="25 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="101">
    <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="107">
    <literal><![CDATA[
      x v (y v ((x v y) ' v z)) = 1
    ]]></literal>
    <justification jstring="[para(81(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="81 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      (x v y) ^ (y v (x v z)) = y v x
    ]]></literal>
    <justification jstring="[para(12(a,1),26(a,1,1))].">
      <j1 rule="para" parents="12 26"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x ^ (x v y) ' = 0
    ]]></literal>
    <justification jstring="[para(19(a,1),27(a,1,2)),demod(92(2)),flip(a)].">
      <j1 rule="para" parents="19 27"/>
      <j2 rule="demod" parents="92"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="136">
    <literal><![CDATA[
      (x ^ y) v ((x v z) ^ y) = (x v z) ^ y
    ]]></literal>
    <justification jstring="[para(27(a,1),31(a,1,2)),demod(12(4))].">
      <j1 rule="para" parents="27 31"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      x ^ (y v x) ' = 0
    ]]></literal>
    <justification jstring="[para(12(a,1),132(a,1,2,1))].">
      <j1 rule="para" parents="12 132"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y)) ') = 0
    ]]></literal>
    <justification jstring="[para(139(a,1),15(a,1)),flip(a)].">
      <j1 rule="para" parents="139 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      x v (y v (z ^ (x ^ u))) = y v x
    ]]></literal>
    <justification jstring="[para(86(a,1),23(a,1,2)),flip(a)].">
      <j1 rule="para" parents="86 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      x v (x ^ y) ' = 1
    ]]></literal>
    <justification jstring="[para(18(a,1),29(a,1,2)),demod(89(2)),flip(a)].">
      <j1 rule="para" parents="18 29"/>
      <j2 rule="demod" parents="89"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="193">
    <literal><![CDATA[
      x v (y ^ x) ' = 1
    ]]></literal>
    <justification jstring="[para(14(a,1),183(a,1,2,1))].">
      <j1 rule="para" parents="14 183"/>
    </justification>
  </clause>

  <clause id="198">
    <literal><![CDATA[
      x v (y v (z ^ (x v y)) ') = 1
    ]]></literal>
    <justification jstring="[para(193(a,1),13(a,1)),flip(a)].">
      <j1 rule="para" parents="193 13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="214">
    <literal><![CDATA[
      x v (y v (y v x) ') = 1
    ]]></literal>
    <justification jstring="[para(12(a,1),36(a,1,2,2,1))].">
      <j1 rule="para" parents="12 36"/>
    </justification>
  </clause>

  <clause id="229">
    <literal><![CDATA[
      x ^ (y ^ (y ^ x) ') = 0
    ]]></literal>
    <justification jstring="[para(14(a,1),39(a,1,2,2,1))].">
      <j1 rule="para" parents="14 39"/>
    </justification>
  </clause>

  <clause id="243">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x ^ z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(14(a,1),32(a,1,1))].">
      <j1 rule="para" parents="14 32"/>
    </justification>
  </clause>

  <clause id="269">
    <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="272">
    <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="317">
    <literal><![CDATA[
      x v (y v z) != 1
    ]]></literal>
    <literal><![CDATA[
      (y v x) ^ z != 0
    ]]></literal>
    <literal><![CDATA[
      (y v x) ' = z
    ]]></literal>
    <justification jstring="[para(23(a,1),42(a,1))].">
      <j1 rule="para" parents="23 42"/>
    </justification>
  </clause>

  <clause id="345">
    <literal><![CDATA[
      x v (y v (z ^ ((x v y) ^ z) ')) != 1
    ]]></literal>
    <literal><![CDATA[
      z ^ ((x v y) ^ z) ' = (x v y) '
    ]]></literal>
    <justification jstring="[para(39(a,1),42(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="39 42"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="514">
    <literal><![CDATA[
      x v (y v (y v (x ^ z)) ') = 1
    ]]></literal>
    <justification jstring="[para(214(a,1),29(a,1,2)),demod(89(2)),flip(a)].">
      <j1 rule="para" parents="214 29"/>
      <j2 rule="demod" parents="89"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="515">
    <literal><![CDATA[
      x ^ (y v (y v x) ') != 0
    ]]></literal>
    <literal><![CDATA[
      (y v (y v x) ') ' = x
    ]]></literal>
    <justification jstring="[para(214(a,1),41(a,1)),demod(14(7)),xx(a)].">
      <j1 rule="para" parents="214 41"/>
      <j2 rule="demod" parents="14"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="524">
    <literal><![CDATA[
      x ^ (y ^ (y ^ (x v z)) ') = 0
    ]]></literal>
    <justification jstring="[para(229(a,1),27(a,1,2)),demod(92(2)),flip(a)].">
      <j1 rule="para" parents="229 27"/>
      <j2 rule="demod" parents="92"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="525">
    <literal><![CDATA[
      x v (y v (z ^ (z ^ (x v y)) ')) != 1
    ]]></literal>
    <literal><![CDATA[
      z ^ (z ^ (x v y)) ' = (x v y) '
    ]]></literal>
    <justification jstring="[para(229(a,1),42(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="229 42"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="794">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z v y)) = x ^ (z v y)
    ]]></literal>
    <justification jstring="[para(100(a,1),31(a,1,2)),demod(12(4))].">
      <j1 rule="para" parents="100 31"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="1130">
    <literal><![CDATA[
      x ^ (x ' v (x ^ y)) = x ^ (x ' v y)
    ]]></literal>
    <justification jstring="[para(19(a,1),52(a,1,2,2,2,1)),demod(71(3))].">
      <j1 rule="para" parents="19 52"/>
      <j2 rule="demod" parents="71"/>
    </justification>
  </clause>

  <clause id="2457">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z)) ' = 1
    ]]></literal>
    <justification jstring="[para(514(a,1),101(a,1)),flip(a)].">
      <j1 rule="para" parents="514 101"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2553">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z)) ' = 0
    ]]></literal>
    <justification jstring="[para(524(a,1),96(a,1)),flip(a)].">
      <j1 rule="para" parents="524 96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3681">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(16(a,1),136(a,1,1))].">
      <j1 rule="para" parents="16 136"/>
    </justification>
  </clause>

  <clause id="7104">
    <literal><![CDATA[
      (x ^ y) v (z v (y ^ (x ^ u))) = z v (y ^ x)
    ]]></literal>
    <justification jstring="[para(243(a,1),23(a,1,2)),flip(a)].">
      <j1 rule="para" parents="243 23"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13555">
    <literal><![CDATA[
      x v ((y v x) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(110(a,1),794(a,1,2)),demod(23(5),12(4),3681(4),110(8))].">
      <j1 rule="para" parents="110 794"/>
      <j2 rule="demod" parents="23 12 3681 110"/>
    </justification>
  </clause>

  <clause id="13559">
    <literal><![CDATA[
      x v ((x v y) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(12(a,1),13555(a,1,2,1))].">
      <j1 rule="para" parents="12 13555"/>
    </justification>
  </clause>

  <clause id="13577">
    <literal><![CDATA[
      x v ((x v y) ^ (z v y)) = x v y
    ]]></literal>
    <justification jstring="[para(12(a,1),13559(a,1,2,2))].">
      <j1 rule="para" parents="12 13559"/>
    </justification>
  </clause>

  <clause id="13770">
    <literal><![CDATA[
      x ^ (x ' v y) = x ^ ((x ^ y) v x ')
    ]]></literal>
    <justification jstring="[para(12(a,1),1130(a,1,2)),flip(a)].">
      <j1 rule="para" parents="12 1130"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13774">
    <literal><![CDATA[
      x ^ (x ' v (x v y) ') = 0
    ]]></literal>
    <justification jstring="[para(132(a,1),1130(a,1,2,2)),demod(12(3),71(3),19(2)),flip(a)].">
      <j1 rule="para" parents="132 1130"/>
      <j2 rule="demod" parents="12 71 19"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14007">
    <literal><![CDATA[
      x ' v (x v y) ' = x '
    ]]></literal>
    <justification jstring="[hyper(317,a,107,a,b,13774,a),demod(34(1),34(2),34(3)),flip(a)].">
      <j1 rule="hyper" parents="317 107 13774"/>
      <j2 rule="demod" parents="34 34 34"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14019">
    <literal><![CDATA[
      x ' v (y v x) ' = x '
    ]]></literal>
    <justification jstring="[para(12(a,1),14007(a,1,2,1))].">
      <j1 rule="para" parents="12 14007"/>
    </justification>
  </clause>

  <clause id="14023">
    <literal><![CDATA[
      x ' ^ (x v y) ' = (x v y) '
    ]]></literal>
    <justification jstring="[para(14007(a,1),25(a,1,2)),demod(14(4))].">
      <j1 rule="para" parents="14007 25"/>
      <j2 rule="demod" parents="14"/>
    </justification>
  </clause>

  <clause id="14750">
    <literal><![CDATA[
      x ' v (x ^ y) ' = (x ^ y) '
    ]]></literal>
    <justification jstring="[para(17(a,1),14019(a,1,2,1)),demod(12(4))].">
      <j1 rule="para" parents="17 14019"/>
      <j2 rule="demod" parents="12"/>
    </justification>
  </clause>

  <clause id="15151">
    <literal><![CDATA[
      x ^ (y v (x ^ y ')) ' = 0
    ]]></literal>
    <justification jstring="[para(14023(a,1),146(a,1,2))].">
      <j1 rule="para" parents="14023 146"/>
    </justification>
  </clause>

  <clause id="15194">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x ^ y) ') = y
    ]]></literal>
    <justification jstring="[hyper(269,a,2457,a,b,15151,a),demod(272(7))].">
      <j1 rule="hyper" parents="269 2457 15151"/>
      <j2 rule="demod" parents="272"/>
    </justification>
  </clause>

  <clause id="15411">
    <literal><![CDATA[
      x ' ^ (x ^ y) ' = x '
    ]]></literal>
    <justification jstring="[para(14750(a,1),16(a,1,2))].">
      <j1 rule="para" parents="14750 16"/>
    </justification>
  </clause>

  <clause id="15415">
    <literal><![CDATA[
      x ' ^ (y ^ (x ^ z) ') = y ^ x '
    ]]></literal>
    <justification jstring="[para(14750(a,1),85(a,1,2,2))].">
      <j1 rule="para" parents="14750 85"/>
    </justification>
  </clause>

  <clause id="15422">
    <literal><![CDATA[
      x v (y ^ (x v y ')) ' = 1
    ]]></literal>
    <justification jstring="[para(14750(a,1),198(a,1,2))].">
      <j1 rule="para" parents="14750 198"/>
    </justification>
  </clause>

  <clause id="15667">
    <literal><![CDATA[
      (x v y) ^ (y v (x v y) ') = y
    ]]></literal>
    <justification jstring="[hyper(269,a,15422,a,b,2553,a),demod(272(7))].">
      <j1 rule="hyper" parents="269 15422 2553"/>
      <j2 rule="demod" parents="272"/>
    </justification>
  </clause>

  <clause id="15685">
    <literal><![CDATA[
      x v (y ' ^ (x v y)) ' = 1
    ]]></literal>
    <justification jstring="[para(272(a,1),15422(a,1,2,1,2,2))].">
      <j1 rule="para" parents="272 15422"/>
    </justification>
  </clause>

  <clause id="15816">
    <literal><![CDATA[
      x v ((x v y) ^ y ') ' = 1
    ]]></literal>
    <justification jstring="[para(14(a,1),15685(a,1,2,1))].">
      <j1 rule="para" parents="14 15685"/>
    </justification>
  </clause>

  <clause id="15850">
    <literal><![CDATA[
      x v ((y v x) ^ y ') ' = 1
    ]]></literal>
    <justification jstring="[para(12(a,1),15816(a,1,2,1,1))].">
      <j1 rule="para" parents="12 15816"/>
    </justification>
  </clause>

  <clause id="15948">
    <literal><![CDATA[
      x v ((x v y) ^ x ') = x v y
    ]]></literal>
    <justification jstring="[para(16(a,1),15194(a,1,1)),demod(16(3))].">
      <j1 rule="para" parents="16 15194"/>
      <j2 rule="demod" parents="16"/>
    </justification>
  </clause>

  <clause id="15967">
    <literal><![CDATA[
      x ' v (x ^ (x ^ y) ') = (x ^ y) '
    ]]></literal>
    <justification jstring="[para(15411(a,1),15194(a,1,1)),demod(15411(7),272(5),14(4))].">
      <j1 rule="para" parents="15411 15194"/>
      <j2 rule="demod" parents="15411 272 14"/>
    </justification>
  </clause>

  <clause id="16047">
    <literal><![CDATA[
      (x v y) ^ (y v (y v x) ') = y
    ]]></literal>
    <justification jstring="[para(12(a,1),15667(a,1,2,2,1))].">
      <j1 rule="para" parents="12 15667"/>
    </justification>
  </clause>

  <clause id="16048">
    <literal><![CDATA[
      x ^ ((x ^ y) v x ') = x ^ y
    ]]></literal>
    <justification jstring="[para(17(a,1),15667(a,1,1)),demod(17(3))].">
      <j1 rule="para" parents="17 15667"/>
      <j2 rule="demod" parents="17"/>
    </justification>
  </clause>

  <clause id="16069">
    <literal><![CDATA[
      x ^ (x ' v y) = x ^ y
    ]]></literal>
    <justification jstring="[back_demod(13770),demod(16048(7))].">
      <j1 rule="back_demod" parents="13770"/>
      <j2 rule="demod" parents="16048"/>
    </justification>
  </clause>

  <clause id="16077">
    <literal><![CDATA[
      x ^ (y v x ') = x ^ y
    ]]></literal>
    <justification jstring="[para(12(a,1),16069(a,1,2))].">
      <j1 rule="para" parents="12 16069"/>
    </justification>
  </clause>

  <clause id="16287">
    <literal><![CDATA[
      x ^ (y v (z v x) ') = x ^ y
    ]]></literal>
    <justification jstring="[para(16077(a,1),96(a,1,2)),demod(96(3)),flip(a)].">
      <j1 rule="para" parents="16077 96"/>
      <j2 rule="demod" parents="96"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16298">
    <literal><![CDATA[
      (x v y ') ' = y ^ (y ^ x) '
    ]]></literal>
    <justification jstring="[para(16077(a,1),525(a,1,2,2,2,1)),demod(15967(5),193(3),16077(6)),flip(b),xx(a)].">
      <j1 rule="para" parents="16077 525"/>
      <j2 rule="demod" parents="15967 193 16077"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="16300">
    <literal><![CDATA[
      (x v y) ^ x ' = y ^ x '
    ]]></literal>
    <justification jstring="[para(15850(a,1),16077(a,1,2)),demod(14(5),69(5),14(7),96(7))].">
      <j1 rule="para" parents="15850 16077"/>
      <j2 rule="demod" parents="14 69 14 96"/>
    </justification>
  </clause>

  <clause id="16303">
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x ^ y ' = x
    ]]></literal>
    <justification jstring="[back_demod(515),demod(16287(4),16298(7),14(6),16(6),16300(6))].">
      <j1 rule="back_demod" parents="515"/>
      <j2 rule="demod" parents="16287 16298 14 16 16300"/>
    </justification>
  </clause>

  <clause id="16312">
    <literal><![CDATA[
      x v (y ^ x ') = x v y
    ]]></literal>
    <justification jstring="[back_demod(15948),demod(16300(3))].">
      <j1 rule="back_demod" parents="15948"/>
      <j2 rule="demod" parents="16300"/>
    </justification>
  </clause>

  <clause id="16320">
    <literal><![CDATA[
      x v (x ' ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(14(a,1),16312(a,1,2))].">
      <j1 rule="para" parents="14 16312"/>
    </justification>
  </clause>

  <clause id="16323">
    <literal><![CDATA[
      x v (y ^ (x ^ z) ') = x v y
    ]]></literal>
    <justification jstring="[para(16312(a,1),29(a,1,2)),demod(29(3)),flip(a)].">
      <j1 rule="para" parents="16312 29"/>
      <j2 rule="demod" parents="29"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16324">
    <literal><![CDATA[
      x ' v (y ^ x) = x ' v y
    ]]></literal>
    <justification jstring="[para(272(a,1),16312(a,1,2,2))].">
      <j1 rule="para" parents="272 16312"/>
    </justification>
  </clause>

  <clause id="16367">
    <literal><![CDATA[
      x ' v (x ^ y) = x ' v y
    ]]></literal>
    <justification jstring="[para(272(a,1),16320(a,1,2,1))].">
      <j1 rule="para" parents="272 16320"/>
    </justification>
  </clause>

  <clause id="16905">
    <literal><![CDATA[
      (x v y) ' = x ' ^ (y ^ x ') '
    ]]></literal>
    <justification jstring="[para(16300(a,1),345(a,1,2,2,2,1)),demod(16323(6),90(3),16300(7)),flip(b),xx(a)].">
      <j1 rule="para" parents="16300 345"/>
      <j2 rule="demod" parents="16323 90 16300"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="17060">
    <literal><![CDATA[
      x ^ (y ^ (y ^ x) ') ' = x
    ]]></literal>
    <justification jstring="[hyper(16303,a,229,a)].">
      <j1 rule="hyper" parents="16303 229"/>
    </justification>
  </clause>

  <clause id="17085">
    <literal><![CDATA[
      x ' v ((y ^ x) v z) = x ' v (y v z)
    ]]></literal>
    <justification jstring="[para(16324(a,1),13(a,1,1)),demod(13(3)),flip(a)].">
      <j1 rule="para" parents="16324 13"/>
      <j2 rule="demod" parents="13"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17137">
    <literal><![CDATA[
      x ' v ((x v y) ^ z) = x ' v z
    ]]></literal>
    <justification jstring="[para(27(a,1),16367(a,1,2)),demod(16367(3)),flip(a)].">
      <j1 rule="para" parents="27 16367"/>
      <j2 rule="demod" parents="16367"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17242">
    <literal><![CDATA[
      x ^ (y ^ x) ' = x ^ y '
    ]]></literal>
    <justification jstring="[para(39(a,1),17060(a,1,2,1,2,1)),demod(82(5),37(5),14(5),15415(5)),flip(a)].">
      <j1 rule="para" parents="39 17060"/>
      <j2 rule="demod" parents="82 37 14 15415"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17276">
    <literal><![CDATA[
      (x v y) ' = x ' ^ y '
    ]]></literal>
    <justification jstring="[back_demod(16905),demod(17242(7))].">
      <j1 rule="back_demod" parents="16905"/>
      <j2 rule="demod" parents="17242"/>
    </justification>
  </clause>

  <clause id="17365">
    <literal><![CDATA[
      (x v y) ^ (y v x ') = y
    ]]></literal>
    <justification jstring="[back_demod(16047),demod(17276(3),16320(5))].">
      <j1 rule="back_demod" parents="16047"/>
      <j2 rule="demod" parents="17276 16320"/>
    </justification>
  </clause>

  <clause id="21604">
    <literal><![CDATA[
      (x v y) ^ (x ' v y) = y
    ]]></literal>
    <justification jstring="[para(12(a,1),17365(a,1,2))].">
      <j1 rule="para" parents="12 17365"/>
    </justification>
  </clause>

  <clause id="22271">
    <literal><![CDATA[
      (x ^ y) v z = (y v z) ^ (y ' v (x v z))
    ]]></literal>
    <justification jstring="[para(101(a,1),21604(a,1,1)),demod(17085(5)),flip(a)].">
      <j1 rule="para" parents="101 21604"/>
      <j2 rule="demod" parents="17085"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22347">
    <literal><![CDATA[
      (x v y) ^ (x ' v (z v y)) = (x v y) ^ (z v y)
    ]]></literal>
    <justification jstring="[para(13577(a,1),21604(a,1,1)),demod(17137(6))].">
      <j1 rule="para" parents="13577 21604"/>
      <j2 rule="demod" parents="17137"/>
    </justification>
  </clause>

  <clause id="22752">
    <literal><![CDATA[
      (x ^ y) v z = (y v z) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_demod(22271),demod(22347(7))].">
      <j1 rule="back_demod" parents="22271"/>
      <j2 rule="demod" parents="22347"/>
    </justification>
  </clause>

  <clause id="23251">
    <literal><![CDATA[
      x v (y ^ z) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_demod(7104),demod(22752(5),75(4),174(5)),flip(a)].">
      <j1 rule="back_demod" parents="7104"/>
      <j2 rule="demod" parents="22752 75 174"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23373">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[back_demod(22),demod(23251(7),12(5),17(5),12(6),23251(6),12(4),12(7),27(9)),xx(a)].">
      <j1 rule="back_demod" parents="22"/>
      <j2 rule="demod" parents="23251 12 17 12 23251 12 12 27"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
