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<!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 September-2006, September 2006.
Process 26878 was started by mccune on cleo.thornwood,
Wed Sep 13 14:38:40 2006
The command was "/home/mccune/LADR/bin/prover9 -f lt.in uc.in H65d.in".
]]></heading>

<proof number="1" length="170" max_count="33">

<comments><![CDATA[
% Proof 1 at 161.07 (+ 1.81) seconds: distributivity.
% Length of proof is 170.
% Level of proof is 29.
% Maximum clause weight is 33.
% Given clauses 1707.
]]></comments>

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

  <clause id="3" type="assumption">
    <literal><![CDATA[
      (x v y) v z = x v (y v z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x ^ y = y ^ x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (x ^ y) ^ z = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      x ^ (x v y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7" type="assumption">
    <literal><![CDATA[
      x v (x ^ y) = x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8" type="assumption">
    <literal><![CDATA[
      x v x' = 1
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="9" type="assumption">
    <literal><![CDATA[
      x ^ x' = 0
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="10" type="assumption">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x' = y
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="11" type="assumption">
    <literal><![CDATA[
      x ^ (y v (z ^ u)) = x ^ (y v (x ^ ((x ^ y) v (z ^ u))))
    ]]></literal>
    <attribute><![CDATA[
      label(H65)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v (x ^ ((x ^ y) v (z ^ u)))) = x ^ (y v (z ^ u))
    ]]></literal>
    <justification jstring="[copy(11),flip(a)].">
      <j1 rule="copy" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13" type="deny">
    <literal><![CDATA[
      (c1 ^ c2) v (c1 ^ c3) != c1 ^ (c2 v c3)
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[deny(1)].">
      <j1 rule="deny" parents="1"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x v (y v z) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(2(a,1),3(a,1,1)),rewrite(3(2))].">
      <j1 rule="para" parents="2 3"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      x ^ (y ^ z) = y ^ (x ^ z)
    ]]></literal>
    <justification jstring="[para(4(a,1),5(a,1,1)),rewrite(5(2))].">
      <j1 rule="para" parents="4 5"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

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

  <clause id="19">
    <literal><![CDATA[
      (x v y) ^ (x v (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(3(a,1),6(a,1,2))].">
      <j1 rule="para" parents="3 6"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x ^ ((x v y) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="6 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      x ^ (y ^ ((x ^ y) v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(6(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="6 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      x v ((x ^ y) v z) = x v z
    ]]></literal>
    <justification jstring="[para(7(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x v (y ^ x) = x
    ]]></literal>
    <justification jstring="[para(4(a,1),7(a,1,2))].">
      <j1 rule="para" parents="4 7"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y ^ z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(5(a,1),7(a,1,2))].">
      <j1 rule="para" parents="5 7"/>
    </justification>
  </clause>

  <clause id="26">
    <literal><![CDATA[
      x ^ x = x
    ]]></literal>
    <justification jstring="[para(7(a,1),6(a,1,2))].">
      <j1 rule="para" parents="7 6"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x v x = x
    ]]></literal>
    <justification jstring="[para(6(a,1),7(a,1,2))].">
      <j1 rule="para" parents="6 7"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      x v (x' v y) = 1 v y
    ]]></literal>
    <justification jstring="[para(8(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="8 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      x v (y v (x v y)') = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="8 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="30">
    <literal><![CDATA[
      x ^ 1 = x
    ]]></literal>
    <justification jstring="[para(8(a,1),6(a,1,2))].">
      <j1 rule="para" parents="8 6"/>
    </justification>
  </clause>

  <clause id="32">
    <literal><![CDATA[
      x ^ (y ^ (x ^ y)') = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="9 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="33">
    <literal><![CDATA[
      x v 0 = x
    ]]></literal>
    <justification jstring="[para(9(a,1),7(a,1,2))].">
      <j1 rule="para" parents="9 7"/>
    </justification>
  </clause>

  <clause id="34">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ x != 0
    ]]></literal>
    <literal><![CDATA[
      y' = x
    ]]></literal>
    <justification jstring="[para(2(a,1),10(a,1))].">
      <j1 rule="para" parents="2 10"/>
    </justification>
  </clause>

  <clause id="35">
    <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(3(a,1),10(a,1))].">
      <j1 rule="para" parents="3 10"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      (x ^ y) v z != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ (y ^ z) != 0
    ]]></literal>
    <literal><![CDATA[
      (x ^ y)' = z
    ]]></literal>
    <justification jstring="[para(5(a,1),10(b,1))].">
      <j1 rule="para" parents="5 10"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x ^ (y v (x ^ ((z ^ u) v (x ^ y)))) = x ^ (y v (z ^ u))
    ]]></literal>
    <justification jstring="[para(2(a,1),12(a,1,2,2,2))].">
      <j1 rule="para" parents="2 12"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      x ^ (y v (x ^ ((x ^ y) v z))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),12(a,1,2,2,2,2)),rewrite(6(7))].">
      <j1 rule="para" parents="6 12"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      x ^ (y ^ x) = y ^ x
    ]]></literal>
    <justification jstring="[para(26(a,1),5(a,2,2)),rewrite(4(2))].">
      <j1 rule="para" parents="26 5"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      x v (x v y) = x v y
    ]]></literal>
    <justification jstring="[para(27(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="27 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      x v (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(27(a,1),3(a,2,2)),rewrite(2(2))].">
      <j1 rule="para" parents="27 3"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="62">
    <literal><![CDATA[
      1 ^ x = x
    ]]></literal>
    <justification jstring="[para(30(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="30 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      0 v x = x
    ]]></literal>
    <justification jstring="[para(33(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="33 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      x v (y v (x ^ z)) = y v x
    ]]></literal>
    <justification jstring="[para(7(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="7 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="69">
    <literal><![CDATA[
      x v (y v x') = y v 1
    ]]></literal>
    <justification jstring="[para(8(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="8 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(62(a,1),6(a,1))].">
      <j1 rule="para" parents="62 6"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite(73(5))].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="73"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      0' = 1
    ]]></literal>
    <justification jstring="[hyper(10,a,64,a,b,30,a)].">
      <j1 rule="hyper" parents="10 64 30"/>
    </justification>
  </clause>

  <clause id="76">
    <literal><![CDATA[
      0 ^ x = 0
    ]]></literal>
    <justification jstring="[para(64(a,1),6(a,1,2))].">
      <j1 rule="para" parents="64 6"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      x ^ (y ^ (x v z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(6(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="6 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x v (y ^ (x ^ z)) = x
    ]]></literal>
    <justification jstring="[para(17(a,1),7(a,1,2))].">
      <j1 rule="para" parents="17 7"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      x ^ (y ^ x') = y ^ 0
    ]]></literal>
    <justification jstring="[para(9(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="9 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="82">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(73(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="73 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      x v (y v x') = 1
    ]]></literal>
    <justification jstring="[back_rewrite(69),rewrite(82(5))].">
      <j1 rule="back_rewrite" parents="69"/>
      <j2 rule="rewrite" parents="82"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(76(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="76 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="86">
    <literal><![CDATA[
      x ^ (y ^ x') = 0
    ]]></literal>
    <justification jstring="[back_rewrite(80),rewrite(85(5))].">
      <j1 rule="back_rewrite" parents="80"/>
      <j2 rule="rewrite" parents="85"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x ^ ((y v x) ^ z) = x ^ z
    ]]></literal>
    <justification jstring="[para(18(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="18 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="93">
    <literal><![CDATA[
      x ^ (y ^ (z v x)) = y ^ x
    ]]></literal>
    <justification jstring="[para(18(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="18 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="94">
    <literal><![CDATA[
      x v ((y ^ x) v z) = x v z
    ]]></literal>
    <justification jstring="[para(24(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="24 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="96">
    <literal><![CDATA[
      x v (y ^ (z ^ x)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),24(a,1,2))].">
      <j1 rule="para" parents="5 24"/>
    </justification>
  </clause>

  <clause id="98">
    <literal><![CDATA[
      x v (y v (z ^ x)) = y v x
    ]]></literal>
    <justification jstring="[para(24(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="24 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="100">
    <literal><![CDATA[
      x v (y v ((x v y)' v z)) = 1
    ]]></literal>
    <justification jstring="[para(74(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="74 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="103">
    <literal><![CDATA[
      (x v y) ^ (y v (x v z)) = y v x
    ]]></literal>
    <justification jstring="[para(2(a,1),19(a,1,1))].">
      <j1 rule="para" parents="2 19"/>
    </justification>
  </clause>

  <clause id="110">
    <literal><![CDATA[
      (x v y) ^ (z ^ (x v (y v u))) = z ^ (x v y)
    ]]></literal>
    <justification jstring="[para(19(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="19 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="125">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),20(a,1,2)),rewrite(85(2)),flip(a)].">
      <j1 rule="para" parents="9 20"/>
      <j2 rule="rewrite" parents="85"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      (x ^ y) v ((x v z) ^ y) = (x v z) ^ y
    ]]></literal>
    <justification jstring="[para(20(a,1),24(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="20 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),125(a,1,2,1))].">
      <j1 rule="para" parents="2 125"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))') = 0
    ]]></literal>
    <justification jstring="[para(132(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="132 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      x ^ (y ^ (z ^ x)) = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[para(5(a,1),54(a,1,2)),rewrite(5(5))].">
      <j1 rule="para" parents="5 54"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="145">
    <literal><![CDATA[
      (x v y) ^ (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(54(a,1),20(a,2)),rewrite(144(4))].">
      <j1 rule="para" parents="54 20"/>
      <j2 rule="rewrite" parents="144"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      x v (y v (x v z)) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(58(a,1),3(a,2,2)),rewrite(15(3),3(2))].">
      <j1 rule="para" parents="58 3"/>
      <j2 rule="rewrite" parents="15 3"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      (x ^ y) v (y ^ ((x ^ y) v z)) = y ^ ((x ^ y) v z)
    ]]></literal>
    <justification jstring="[para(21(a,1),24(a,1,2)),rewrite(2(5))].">
      <j1 rule="para" parents="21 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      (x v y) ^ (y v x) = x v y
    ]]></literal>
    <justification jstring="[para(60(a,1),19(a,1,2))].">
      <j1 rule="para" parents="60 19"/>
    </justification>
  </clause>

  <clause id="165">
    <literal><![CDATA[
      x v (y ^ (z ^ (x ^ u))) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),79(a,1,2))].">
      <j1 rule="para" parents="5 79"/>
    </justification>
  </clause>

  <clause id="167">
    <literal><![CDATA[
      x v (y v (z ^ (x ^ u))) = y v x
    ]]></literal>
    <justification jstring="[para(79(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="79 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      x v (y v (((x v y) ^ z) v u)) = x v (y v u)
    ]]></literal>
    <justification jstring="[para(22(a,1),3(a,1)),rewrite(3(2)),flip(a)].">
      <j1 rule="para" parents="22 3"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      (x ^ y) v ((x ^ (y ^ z)) v u) = (x ^ y) v u
    ]]></literal>
    <justification jstring="[para(5(a,1),22(a,1,2,1))].">
      <j1 rule="para" parents="5 22"/>
    </justification>
  </clause>

  <clause id="176">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),22(a,1,2)),rewrite(82(2)),flip(a)].">
      <j1 rule="para" parents="8 22"/>
      <j2 rule="rewrite" parents="82"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="186">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),176(a,1,2,1))].">
      <j1 rule="para" parents="4 176"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      x v (y v (z ^ (x v y))') = 1
    ]]></literal>
    <justification jstring="[para(186(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="186 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      x v (y v (y v x)') = 1
    ]]></literal>
    <justification jstring="[para(2(a,1),29(a,1,2,2,1))].">
      <j1 rule="para" parents="2 29"/>
    </justification>
  </clause>

  <clause id="222">
    <literal><![CDATA[
      x ^ (y ^ (y ^ x)') = 0
    ]]></literal>
    <justification jstring="[para(4(a,1),32(a,1,2,2,1))].">
      <j1 rule="para" parents="4 32"/>
    </justification>
  </clause>

  <clause id="237">
    <literal><![CDATA[
      (x ^ y) v (y ^ (z ^ x)) = x ^ y
    ]]></literal>
    <justification jstring="[para(4(a,1),25(a,1,2)),rewrite(5(3))].">
      <j1 rule="para" parents="4 25"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="238">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (x ^ (y ^ (z ^ u))) = x ^ (y ^ z)
    ]]></literal>
    <justification jstring="[para(5(a,1),25(a,1,1)),rewrite(5(5),5(8))].">
      <j1 rule="para" parents="5 25"/>
      <j2 rule="rewrite" parents="5 5"/>
    </justification>
  </clause>

  <clause id="262">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      y' = x
    ]]></literal>
    <justification jstring="[para(4(a,1),34(b,1))].">
      <j1 rule="para" parents="4 34"/>
    </justification>
  </clause>

  <clause id="265">
    <literal><![CDATA[
      x'' = x
    ]]></literal>
    <justification jstring="[para(8(a,1),34(a,1)),rewrite(4(5),9(5)),xx(a),xx(b)].">
      <j1 rule="para" parents="8 34"/>
      <j2 rule="rewrite" parents="4 9"/>
      <j2 rule="xx"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="284">
    <literal><![CDATA[
      x' ^ (y ^ x) = 0
    ]]></literal>
    <justification jstring="[para(265(a,1),86(a,1,2,2))].">
      <j1 rule="para" parents="265 86"/>
    </justification>
  </clause>

  <clause id="310">
    <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(15(a,1),35(a,1))].">
      <j1 rule="para" parents="15 35"/>
    </justification>
  </clause>

  <clause id="338">
    <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(32(a,1),35(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="32 35"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="507">
    <literal><![CDATA[
      x v (y v (y v (x ^ z))') = 1
    ]]></literal>
    <justification jstring="[para(207(a,1),22(a,1,2)),rewrite(82(2)),flip(a)].">
      <j1 rule="para" parents="207 22"/>
      <j2 rule="rewrite" parents="82"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="508">
    <literal><![CDATA[
      x ^ (y v (y v x)') != 0
    ]]></literal>
    <literal><![CDATA[
      (y v (y v x)')' = x
    ]]></literal>
    <justification jstring="[para(207(a,1),34(a,1)),rewrite(4(7)),xx(a)].">
      <j1 rule="para" parents="207 34"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="510">
    <literal><![CDATA[
      x ^ (y ^ (z v (z v (x ^ y))')) != 0
    ]]></literal>
    <literal><![CDATA[
      z v (z v (x ^ y))' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(207(a,1),37(a,1)),flip(c),xx(a)].">
      <j1 rule="para" parents="207 37"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="517">
    <literal><![CDATA[
      x ^ (y ^ (y ^ (x v z))') = 0
    ]]></literal>
    <justification jstring="[para(222(a,1),20(a,1,2)),rewrite(85(2)),flip(a)].">
      <j1 rule="para" parents="222 20"/>
      <j2 rule="rewrite" parents="85"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="518">
    <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(222(a,1),35(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="222 35"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="532">
    <literal><![CDATA[
      x ^ (y v (x ^ (z v (x ^ y)))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),40(a,1,2,2,2,1)),rewrite(6(7))].">
      <j1 rule="para" parents="6 40"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="536">
    <literal><![CDATA[
      x v ((y ^ (x v (z ^ u))) v (y ^ ((z ^ u) v (y ^ x)))) = x v (y ^ ((z ^ u) v (y ^ x)))
    ]]></literal>
    <justification jstring="[para(40(a,1),24(a,1,2)),rewrite(2(9),15(9))].">
      <j1 rule="para" parents="40 24"/>
      <j2 rule="rewrite" parents="2 15"/>
    </justification>
  </clause>

  <clause id="549">
    <literal><![CDATA[
      x v ((y ^ ((z ^ u) v (y ^ x))) v (v ^ (y ^ (x v (z ^ u))))) = x v (y ^ ((z ^ u) v (y ^ x)))
    ]]></literal>
    <justification jstring="[para(40(a,1),96(a,1,2,2)),rewrite(3(10))].">
      <j1 rule="para" parents="40 96"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="668">
    <literal><![CDATA[
      x v (y v (z v ((x v y) ^ u))) = z v (x v y)
    ]]></literal>
    <justification jstring="[para(68(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="68 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="787">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z v y)) = x ^ (z v y)
    ]]></literal>
    <justification jstring="[para(93(a,1),24(a,1,2)),rewrite(2(4))].">
      <j1 rule="para" parents="93 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="943">
    <literal><![CDATA[
      (x v (y v z)) ^ (u ^ (x v y)) = u ^ (x v y)
    ]]></literal>
    <justification jstring="[para(3(a,1),145(a,1,1))].">
      <j1 rule="para" parents="3 145"/>
    </justification>
  </clause>

  <clause id="974">
    <literal><![CDATA[
      (x v (y v z)) ^ (y v x) = y v x
    ]]></literal>
    <justification jstring="[para(157(a,1),145(a,1,2)),rewrite(3(2),157(7))].">
      <j1 rule="para" parents="157 145"/>
      <j2 rule="rewrite" parents="3 157"/>
    </justification>
  </clause>

  <clause id="1075">
    <literal><![CDATA[
      x v (y v (z ^ (u ^ (x ^ v)))) = y v x
    ]]></literal>
    <justification jstring="[para(165(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="165 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1123">
    <literal><![CDATA[
      x ^ (x' v (x ^ y)) = x ^ (x' v y)
    ]]></literal>
    <justification jstring="[para(9(a,1),45(a,1,2,2,2,1)),rewrite(64(3))].">
      <j1 rule="para" parents="9 45"/>
      <j2 rule="rewrite" parents="64"/>
    </justification>
  </clause>

  <clause id="2450">
    <literal><![CDATA[
      x v ((y ^ x) v (x ^ z))' = 1
    ]]></literal>
    <justification jstring="[para(507(a,1),94(a,1)),flip(a)].">
      <j1 rule="para" parents="507 94"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2546">
    <literal><![CDATA[
      x ^ ((y v x) ^ (x v z))' = 0
    ]]></literal>
    <justification jstring="[para(517(a,1),89(a,1)),flip(a)].">
      <j1 rule="para" parents="517 89"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3674">
    <literal><![CDATA[
      x v ((x v y) ^ (x v z)) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[para(6(a,1),129(a,1,1))].">
      <j1 rule="para" parents="6 129"/>
    </justification>
  </clause>

  <clause id="7204">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (z ^ x) = z ^ x
    ]]></literal>
    <justification jstring="[para(237(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="237 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7677">
    <literal><![CDATA[
      (x v (y v z)) ^ (u ^ (y v x)) = u ^ (y v x)
    ]]></literal>
    <justification jstring="[para(974(a,1),17(a,1,2)),flip(a)].">
      <j1 rule="para" parents="974 17"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8725">
    <literal><![CDATA[
      (x ^ (y ^ z)) v ((x ^ (y ^ (z ^ u))) v v) = (x ^ (y ^ z)) v v
    ]]></literal>
    <justification jstring="[para(5(a,1),175(a,1,1)),rewrite(5(5),5(9))].">
      <j1 rule="para" parents="5 175"/>
      <j2 rule="rewrite" parents="5 5"/>
    </justification>
  </clause>

  <clause id="9593">
    <literal><![CDATA[
      (x ^ (y ^ z)) v ((z ^ x) v u) = (z ^ x) v u
    ]]></literal>
    <justification jstring="[para(7204(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="7204 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="9641">
    <literal><![CDATA[
      x v ((y ^ z) v ((x v (z ^ (u ^ y))) ^ v)) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(7204(a,1),174(a,2,2)),rewrite(2(8),9593(9))].">
      <j1 rule="para" parents="7204 174"/>
      <j2 rule="rewrite" parents="2 9593"/>
    </justification>
  </clause>

  <clause id="25225">
    <literal><![CDATA[
      x ^ ((y ^ z) v (x ^ y)) = x ^ y
    ]]></literal>
    <justification jstring="[para(79(a,1),532(a,1,2,2,2)),rewrite(2(6),7(6))].">
      <j1 rule="para" parents="79 532"/>
      <j2 rule="rewrite" parents="2 7"/>
    </justification>
  </clause>

  <clause id="25456">
    <literal><![CDATA[
      x v ((y v x) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(103(a,1),787(a,1,2)),rewrite(15(5),2(4),3674(4),103(8))].">
      <j1 rule="para" parents="103 787"/>
      <j2 rule="rewrite" parents="15 2 3674 103"/>
    </justification>
  </clause>

  <clause id="25460">
    <literal><![CDATA[
      x v ((x v y) ^ (y v z)) = x v y
    ]]></literal>
    <justification jstring="[para(2(a,1),25456(a,1,2,1))].">
      <j1 rule="para" parents="2 25456"/>
    </justification>
  </clause>

  <clause id="25478">
    <literal><![CDATA[
      x v ((x v y) ^ (z v y)) = x v y
    ]]></literal>
    <justification jstring="[para(2(a,1),25460(a,1,2,2))].">
      <j1 rule="para" parents="2 25460"/>
    </justification>
  </clause>

  <clause id="25490">
    <literal><![CDATA[
      x v (y ^ (x v (y ^ z))) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(25225(a,1),25460(a,1,2)),rewrite(4(3))].">
      <j1 rule="para" parents="25225 25460"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="25518">
    <literal><![CDATA[
      x v (y v ((x v z) ^ (u v z))) = y v (x v z)
    ]]></literal>
    <justification jstring="[para(25478(a,1),15(a,1,2)),flip(a)].">
      <j1 rule="para" parents="25478 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25671">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ ((x ^ y) v x')
    ]]></literal>
    <justification jstring="[para(2(a,1),1123(a,1,2)),flip(a)].">
      <j1 rule="para" parents="2 1123"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25675">
    <literal><![CDATA[
      x ^ (x' v (x v y)') = 0
    ]]></literal>
    <justification jstring="[para(125(a,1),1123(a,1,2,2)),rewrite(2(3),64(3),9(2)),flip(a)].">
      <j1 rule="para" parents="125 1123"/>
      <j2 rule="rewrite" parents="2 64 9"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25910">
    <literal><![CDATA[
      x' v (x v y)' = x'
    ]]></literal>
    <justification jstring="[hyper(310,a,100,a,b,25675,a),rewrite(27(1),27(2),27(3)),flip(a)].">
      <j1 rule="hyper" parents="310 100 25675"/>
      <j2 rule="rewrite" parents="27 27 27"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25922">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(2(a,1),25910(a,1,2,1))].">
      <j1 rule="para" parents="2 25910"/>
    </justification>
  </clause>

  <clause id="25926">
    <literal><![CDATA[
      x' ^ (x v y)' = (x v y)'
    ]]></literal>
    <justification jstring="[para(25910(a,1),18(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="25910 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="26673">
    <literal><![CDATA[
      x' v (x ^ y)' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(7(a,1),25922(a,1,2,1)),rewrite(2(4))].">
      <j1 rule="para" parents="7 25922"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="27341">
    <literal><![CDATA[
      x ^ (y v (x ^ y'))' = 0
    ]]></literal>
    <justification jstring="[para(25926(a,1),139(a,1,2))].">
      <j1 rule="para" parents="25926 139"/>
    </justification>
  </clause>

  <clause id="27552">
    <literal><![CDATA[
      (x ^ y) v (y ^ (x ^ y)') = y
    ]]></literal>
    <justification jstring="[hyper(262,a,2450,a,b,27341,a),rewrite(265(7))].">
      <j1 rule="hyper" parents="262 2450 27341"/>
      <j2 rule="rewrite" parents="265"/>
    </justification>
  </clause>

  <clause id="28174">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(26673(a,1),6(a,1,2))].">
      <j1 rule="para" parents="26673 6"/>
    </justification>
  </clause>

  <clause id="28181">
    <literal><![CDATA[
      x' ^ (y ^ (x ^ z)') = y ^ x'
    ]]></literal>
    <justification jstring="[para(26673(a,1),78(a,1,2,2))].">
      <j1 rule="para" parents="26673 78"/>
    </justification>
  </clause>

  <clause id="28188">
    <literal><![CDATA[
      x v (y ^ (x v y'))' = 1
    ]]></literal>
    <justification jstring="[para(26673(a,1),191(a,1,2))].">
      <j1 rule="para" parents="26673 191"/>
    </justification>
  </clause>

  <clause id="28496">
    <literal><![CDATA[
      x ^ (x' ^ y)' = x
    ]]></literal>
    <justification jstring="[para(265(a,1),28174(a,1,1)),rewrite(265(6))].">
      <j1 rule="para" parents="265 28174"/>
      <j2 rule="rewrite" parents="265"/>
    </justification>
  </clause>

  <clause id="29347">
    <literal><![CDATA[
      (x v y) ^ (y v (x v y)') = y
    ]]></literal>
    <justification jstring="[hyper(262,a,28188,a,b,2546,a),rewrite(265(7))].">
      <j1 rule="hyper" parents="262 28188 2546"/>
      <j2 rule="rewrite" parents="265"/>
    </justification>
  </clause>

  <clause id="29351">
    <literal><![CDATA[
      x v (y' ^ (x v y))' = 1
    ]]></literal>
    <justification jstring="[para(265(a,1),28188(a,1,2,1,2,2))].">
      <j1 rule="para" parents="265 28188"/>
    </justification>
  </clause>

  <clause id="29757">
    <literal><![CDATA[
      x v ((x v y) ^ y')' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),29351(a,1,2,1))].">
      <j1 rule="para" parents="4 29351"/>
    </justification>
  </clause>

  <clause id="29829">
    <literal><![CDATA[
      x v ((y v x) ^ y')' = 1
    ]]></literal>
    <justification jstring="[para(2(a,1),29757(a,1,2,1,1))].">
      <j1 rule="para" parents="2 29757"/>
    </justification>
  </clause>

  <clause id="30361">
    <literal><![CDATA[
      x v ((x v y) ^ x') = x v y
    ]]></literal>
    <justification jstring="[para(6(a,1),27552(a,1,1)),rewrite(6(3))].">
      <j1 rule="para" parents="6 27552"/>
      <j2 rule="rewrite" parents="6"/>
    </justification>
  </clause>

  <clause id="30396">
    <literal><![CDATA[
      x' v (x ^ (x ^ y)') = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(28174(a,1),27552(a,1,1)),rewrite(28174(7),265(5),4(4))].">
      <j1 rule="para" parents="28174 27552"/>
      <j2 rule="rewrite" parents="28174 265 4"/>
    </justification>
  </clause>

  <clause id="30861">
    <literal><![CDATA[
      x ^ ((x ^ y) v x') = x ^ y
    ]]></literal>
    <justification jstring="[para(7(a,1),29347(a,1,1)),rewrite(7(3))].">
      <j1 rule="para" parents="7 29347"/>
      <j2 rule="rewrite" parents="7"/>
    </justification>
  </clause>

  <clause id="30900">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ y
    ]]></literal>
    <justification jstring="[back_rewrite(25671),rewrite(30861(7))].">
      <j1 rule="back_rewrite" parents="25671"/>
      <j2 rule="rewrite" parents="30861"/>
    </justification>
  </clause>

  <clause id="30907">
    <literal><![CDATA[
      x ^ (y v x') = x ^ y
    ]]></literal>
    <justification jstring="[para(2(a,1),30900(a,1,2))].">
      <j1 rule="para" parents="2 30900"/>
    </justification>
  </clause>

  <clause id="31319">
    <literal><![CDATA[
      x' ^ (y v x) = x' ^ y
    ]]></literal>
    <justification jstring="[para(265(a,1),30907(a,1,2,2))].">
      <j1 rule="para" parents="265 30907"/>
    </justification>
  </clause>

  <clause id="31320">
    <literal><![CDATA[
      x ^ (y v (z v x)') = x ^ y
    ]]></literal>
    <justification jstring="[para(30907(a,1),89(a,1,2)),rewrite(89(3)),flip(a)].">
      <j1 rule="para" parents="30907 89"/>
      <j2 rule="rewrite" parents="89"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31373">
    <literal><![CDATA[
      (x v y')' = y ^ (y ^ x)'
    ]]></literal>
    <justification jstring="[para(30907(a,1),518(a,1,2,2,2,1)),rewrite(30396(5),186(3),30907(6)),flip(b),xx(a)].">
      <j1 rule="para" parents="30907 518"/>
      <j2 rule="rewrite" parents="30396 186 30907"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="31386">
    <literal><![CDATA[
      (x v y) ^ x' = y ^ x'
    ]]></literal>
    <justification jstring="[para(29829(a,1),30907(a,1,2)),rewrite(4(5),62(5),4(7),89(7))].">
      <j1 rule="para" parents="29829 30907"/>
      <j2 rule="rewrite" parents="4 62 4 89"/>
    </justification>
  </clause>

  <clause id="31395">
    <literal><![CDATA[
      x ^ y != 0
    ]]></literal>
    <literal><![CDATA[
      x ^ y' = x
    ]]></literal>
    <justification jstring="[back_rewrite(508),rewrite(31320(4),31373(7),4(6),6(6),31386(6))].">
      <j1 rule="back_rewrite" parents="508"/>
      <j2 rule="rewrite" parents="31320 31373 4 6 31386"/>
    </justification>
  </clause>

  <clause id="31420">
    <literal><![CDATA[
      x v (y ^ x') = x v y
    ]]></literal>
    <justification jstring="[back_rewrite(30361),rewrite(31386(3))].">
      <j1 rule="back_rewrite" parents="30361"/>
      <j2 rule="rewrite" parents="31386"/>
    </justification>
  </clause>

  <clause id="31428">
    <literal><![CDATA[
      x v (x' ^ y) = x v y
    ]]></literal>
    <justification jstring="[para(4(a,1),31420(a,1,2))].">
      <j1 rule="para" parents="4 31420"/>
    </justification>
  </clause>

  <clause id="31429">
    <literal><![CDATA[
      x v (y ^ (z ^ x')) = x v (y ^ z)
    ]]></literal>
    <justification jstring="[para(5(a,1),31420(a,1,2))].">
      <j1 rule="para" parents="5 31420"/>
    </justification>
  </clause>

  <clause id="31431">
    <literal><![CDATA[
      x v (y ^ (x ^ z)') = x v y
    ]]></literal>
    <justification jstring="[para(31420(a,1),22(a,1,2)),rewrite(22(3)),flip(a)].">
      <j1 rule="para" parents="31420 22"/>
      <j2 rule="rewrite" parents="22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="31432">
    <literal><![CDATA[
      x' v (y ^ x) = x' v y
    ]]></literal>
    <justification jstring="[para(265(a,1),31420(a,1,2,2))].">
      <j1 rule="para" parents="265 31420"/>
    </justification>
  </clause>

  <clause id="31444">
    <literal><![CDATA[
      (x ^ y) v (y ^ z) = y ^ ((x ^ y) v z)
    ]]></literal>
    <justification jstring="[para(153(a,1),31420(a,2)),rewrite(5(7),31386(6),31429(6))].">
      <j1 rule="para" parents="153 31420"/>
      <j2 rule="rewrite" parents="5 31386 31429"/>
    </justification>
  </clause>

  <clause id="31529">
    <literal><![CDATA[
      x' v (x ^ y) = x' v y
    ]]></literal>
    <justification jstring="[para(265(a,1),31428(a,1,2,1))].">
      <j1 rule="para" parents="265 31428"/>
    </justification>
  </clause>

  <clause id="31773">
    <literal><![CDATA[
      (x' ^ y)' = x v (x v y)'
    ]]></literal>
    <justification jstring="[para(31428(a,1),510(a,1,2,2,2,1)),rewrite(31320(5),284(3),31428(6)),flip(b),xx(a)].">
      <j1 rule="para" parents="31428 510"/>
      <j2 rule="rewrite" parents="31320 284 31428"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="32095">
    <literal><![CDATA[
      (x v y) ^ (x' ^ y)' = x
    ]]></literal>
    <justification jstring="[para(31428(a,1),31319(a,1,2)),rewrite(4(5),4(9),28496(9))].">
      <j1 rule="para" parents="31428 31319"/>
      <j2 rule="rewrite" parents="4 4 28496"/>
    </justification>
  </clause>

  <clause id="32186">
    <literal><![CDATA[
      (x v y)' = x' ^ (y ^ x')'
    ]]></literal>
    <justification jstring="[para(31386(a,1),338(a,1,2,2,2,1)),rewrite(31431(6),83(3),31386(7)),flip(b),xx(a)].">
      <j1 rule="para" parents="31386 338"/>
      <j2 rule="rewrite" parents="31431 83 31386"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="32368">
    <literal><![CDATA[
      x ^ (y ^ (y ^ x)')' = x
    ]]></literal>
    <justification jstring="[hyper(31395,a,222,a)].">
      <j1 rule="hyper" parents="31395 222"/>
    </justification>
  </clause>

  <clause id="32412">
    <literal><![CDATA[
      x' v ((y ^ x) v z) = x' v (y v z)
    ]]></literal>
    <justification jstring="[para(31432(a,1),3(a,1,1)),rewrite(3(3)),flip(a)].">
      <j1 rule="para" parents="31432 3"/>
      <j2 rule="rewrite" parents="3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32512">
    <literal><![CDATA[
      x' v ((x v y) ^ z) = x' v z
    ]]></literal>
    <justification jstring="[para(20(a,1),31529(a,1,2)),rewrite(31529(3)),flip(a)].">
      <j1 rule="para" parents="20 31529"/>
      <j2 rule="rewrite" parents="31529"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32781">
    <literal><![CDATA[
      (x v y) ^ (y' ^ x)' = y
    ]]></literal>
    <justification jstring="[para(2(a,1),32095(a,1,1))].">
      <j1 rule="para" parents="2 32095"/>
    </justification>
  </clause>

  <clause id="32810">
    <literal><![CDATA[
      x ^ (y ^ x)' = x ^ y'
    ]]></literal>
    <justification jstring="[para(32(a,1),32368(a,1,2,1,2,1)),rewrite(75(5),30(5),4(5),28181(5)),flip(a)].">
      <j1 rule="para" parents="32 32368"/>
      <j2 rule="rewrite" parents="75 30 4 28181"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="32852">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[back_rewrite(32186),rewrite(32810(7))].">
      <j1 rule="back_rewrite" parents="32186"/>
      <j2 rule="rewrite" parents="32810"/>
    </justification>
  </clause>

  <clause id="33036">
    <literal><![CDATA[
      (x' ^ y)' = x v y'
    ]]></literal>
    <justification jstring="[back_rewrite(31773),rewrite(32852(5),31428(7))].">
      <j1 rule="back_rewrite" parents="31773"/>
      <j2 rule="rewrite" parents="32852 31428"/>
    </justification>
  </clause>

  <clause id="36023">
    <literal><![CDATA[
      (x v y) ^ (y v x') = y
    ]]></literal>
    <justification jstring="[back_rewrite(32781),rewrite(33036(4))].">
      <j1 rule="back_rewrite" parents="32781"/>
      <j2 rule="rewrite" parents="33036"/>
    </justification>
  </clause>

  <clause id="40709">
    <literal><![CDATA[
      (x v y) ^ (x' v y) = y
    ]]></literal>
    <justification jstring="[para(2(a,1),36023(a,1,2))].">
      <j1 rule="para" parents="2 36023"/>
    </justification>
  </clause>

  <clause id="40783">
    <literal><![CDATA[
      x v ((y ^ (x v z)) v (y ^ (z v (y ^ x)))) = x v (y ^ (z v (y ^ x)))
    ]]></literal>
    <justification jstring="[para(36023(a,1),536(a,1,2,1,2,2)),rewrite(36023(6),36023(11))].">
      <j1 rule="para" parents="36023 536"/>
      <j2 rule="rewrite" parents="36023 36023"/>
    </justification>
  </clause>

  <clause id="40791">
    <literal><![CDATA[
      x v ((y ^ (z v (y ^ x))) v (u ^ (y ^ (x v z)))) = x v (y ^ (z v (y ^ x)))
    ]]></literal>
    <justification jstring="[para(36023(a,1),549(a,1,2,1,2,1)),rewrite(36023(7),36023(12))].">
      <j1 rule="para" parents="36023 549"/>
      <j2 rule="rewrite" parents="36023 36023"/>
    </justification>
  </clause>

  <clause id="41539">
    <literal><![CDATA[
      (x ^ (y ^ z)) v (u v (x ^ (y ^ (z ^ v)))) = u v (x ^ (y ^ z))
    ]]></literal>
    <justification jstring="[para(238(a,1),668(a,1,2,2,2,1)),rewrite(5(8),5(7),8725(11),238(13))].">
      <j1 rule="para" parents="238 668"/>
      <j2 rule="rewrite" parents="5 5 8725 238"/>
    </justification>
  </clause>

  <clause id="41633">
    <literal><![CDATA[
      (x ^ y) v z = (y v z) ^ (y' v (x v z))
    ]]></literal>
    <justification jstring="[para(94(a,1),40709(a,1,1)),rewrite(32412(5)),flip(a)].">
      <j1 rule="para" parents="94 40709"/>
      <j2 rule="rewrite" parents="32412"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41712">
    <literal><![CDATA[
      (x v y) ^ (x' v (z v y)) = (x v y) ^ (z v y)
    ]]></literal>
    <justification jstring="[para(25478(a,1),40709(a,1,1)),rewrite(32512(6))].">
      <j1 rule="para" parents="25478 40709"/>
      <j2 rule="rewrite" parents="32512"/>
    </justification>
  </clause>

  <clause id="42508">
    <literal><![CDATA[
      (x ^ y) v z = (y v z) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(41633),rewrite(41712(7))].">
      <j1 rule="back_rewrite" parents="41633"/>
      <j2 rule="rewrite" parents="41712"/>
    </justification>
  </clause>

  <clause id="42664">
    <literal><![CDATA[
      x v (y ^ (z ^ u)) = (x v u) ^ ((x v z) ^ (x v y))
    ]]></literal>
    <justification jstring="[back_rewrite(41539),rewrite(42508(7),42508(6),1075(5),167(6),68(8),5(5)),flip(a)].">
      <j1 rule="back_rewrite" parents="41539"/>
      <j2 rule="rewrite" parents="42508 42508 1075 167 68 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42832">
    <literal><![CDATA[
      x v (y ^ (z v (y ^ x))) = (x v y) ^ (z v x)
    ]]></literal>
    <justification jstring="[back_rewrite(40791),rewrite(42664(7),15(5),2(4),25490(4),98(3),2(5),7(5),42508(5),3(4),42508(3),4(7),20(8),42664(8),25518(5),60(5),943(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="40791"/>
      <j2 rule="rewrite" parents="42664 15 2 25490 98 2 7 42508 3 42508 4 20 42664 25518 60 943"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="42836">
    <literal><![CDATA[
      x v ((y v x) ^ z) = (x v z) ^ (y v x)
    ]]></literal>
    <justification jstring="[back_rewrite(40783),rewrite(42508(6),3(5),25490(4),98(3),7(5),42832(7))].">
      <j1 rule="back_rewrite" parents="40783"/>
      <j2 rule="rewrite" parents="42508 3 25490 98 7 42832"/>
    </justification>
  </clause>

  <clause id="43357">
    <literal><![CDATA[
      x ^ (y v (x ^ z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(31444),rewrite(42508(3),7(2),42508(5),20(7))].">
      <j1 rule="back_rewrite" parents="31444"/>
      <j2 rule="rewrite" parents="42508 7 42508 20"/>
    </justification>
  </clause>

  <clause id="43581">
    <literal><![CDATA[
      x v (y ^ z) = (x v y) ^ (x v z)
    ]]></literal>
    <justification jstring="[back_rewrite(9641),rewrite(42664(4),5(7),5(6),42664(8),42508(4),42836(3),5(7),43357(6),42508(8),42508(13),60(14),4(14),18(14),5(12),974(11),5(10),5(9),42664(11),42664(7),58(2),146(4),110(7),58(6),19(8),5(6),7677(6)),flip(a)].">
      <j1 rule="back_rewrite" parents="9641"/>
      <j2 rule="rewrite" parents="42664 5 5 42664 42508 42836 5 43357 42508 42508 60 4 18 5 974 5 5 42664 42664 58 146 110 58 19 5 7677"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="43836">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[back_rewrite(13),rewrite(43581(7),2(5),7(5),2(6),43581(6),2(4),2(7),20(9)),xx(a)].">
      <j1 rule="back_rewrite" parents="13"/>
      <j2 rule="rewrite" parents="43581 2 7 2 43581 2 2 20"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
