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<!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 22-May-2007, May 2007.
Process 27627 was started by mccune on cleo,
Tue May 22 14:49:40 2007
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H27d.in".
]]></heading>

<proof number="1" length="160" max_count="25">

<comments><![CDATA[
% Proof 1 at 21.97 (+ 0.14) seconds: distributivity.
% Length of proof is 160.
% Level of proof is 29.
% Maximum clause weight is 23.
% Given clauses 783.
]]></comments>

  <clause id="1" type="goal">
    <literal><![CDATA[
      x ^ (y v z) = (x ^ y) v (x ^ z)
    ]]></literal>
    <attribute><![CDATA[
      label(distributivity)
    ]]></attribute>
    <attribute><![CDATA[
      label(non_clause)
    ]]></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 ^ (x v u))) = x ^ (y v ((x ^ z) v (z ^ (y v u))))
    ]]></literal>
    <attribute><![CDATA[
      label(H49)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      x ^ (y v ((x ^ z) v (z ^ (y v u)))) = x ^ (y v (z ^ (x v 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[
      label(distributivity)
    ]]></attribute>
    <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="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="23">
    <literal><![CDATA[
      x v (y v ((x v y) ^ z)) = x v y
    ]]></literal>
    <justification jstring="[para(7(a,1),3(a,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="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="31">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0 ^ y
    ]]></literal>
    <justification jstring="[para(9(a,1),5(a,1,1)),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="40">
    <literal><![CDATA[
      x ^ (y v ((x ^ z) v ((y v u) ^ z))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(4(a,1),12(a,1,2,2,2))].">
      <j1 rule="para" parents="4 12"/>
    </justification>
  </clause>

  <clause id="43">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v y'))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(8(a,1),12(a,1,2,2,2,2)),rewrite([30(3),2(2),24(2)]),flip(a)].">
      <j1 rule="para" parents="8 12"/>
      <j2 rule="rewrite" parents="30 2 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="46">
    <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="47">
    <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="51">
    <literal><![CDATA[
      x ^ (x ^ y) = x ^ y
    ]]></literal>
    <justification jstring="[para(26(a,1),5(a,1,1)),flip(a)].">
      <j1 rule="para" parents="26 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="53">
    <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="56">
    <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 ^ (y v ((x ^ z) v (z ^ y))) = x ^ (y v (z ^ (x v y)))
    ]]></literal>
    <justification jstring="[para(27(a,1),12(a,1,2,2,2,2))].">
      <j1 rule="para" parents="27 12"/>
    </justification>
  </clause>

  <clause id="63">
    <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="66">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v y))) = x ^ (y v (z ^ x))
    ]]></literal>
    <justification jstring="[para(33(a,1),12(a,1,2,2,2,2)),rewrite([60(5),33(6)])].">
      <j1 rule="para" parents="33 12"/>
      <j2 rule="rewrite" parents="60 33"/>
    </justification>
  </clause>

  <clause id="70">
    <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="72">
    <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="73">
    <literal><![CDATA[
      1' = 0
    ]]></literal>
    <justification jstring="[hyper(10,a,33,a,b,63,a)].">
      <j1 rule="hyper" parents="10 33 63"/>
    </justification>
  </clause>

  <clause id="74">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(63(a,1),6(a,1))].">
      <j1 rule="para" parents="63 6"/>
    </justification>
  </clause>

  <clause id="75">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite([74(5)])].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="74"/>
    </justification>
  </clause>

  <clause id="76">
    <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="77">
    <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 ^ ((x ^ y) v (y ^ z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(64(a,1),12(a,1,2,2,2,2)),rewrite([64(5),64(8),70(7)])].">
      <j1 rule="para" parents="64 12"/>
      <j2 rule="rewrite" parents="64 64 70"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(31),rewrite([77(5)])].">
      <j1 rule="back_rewrite" parents="31"/>
      <j2 rule="rewrite" parents="77"/>
    </justification>
  </clause>

  <clause id="81">
    <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="85">
    <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="86">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(18(a,1),63(a,1)),flip(a)].">
      <j1 rule="para" parents="18 63"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="87">
    <literal><![CDATA[
      x v (y v x') = 1
    ]]></literal>
    <justification jstring="[back_rewrite(47),rewrite([86(5)])].">
      <j1 rule="back_rewrite" parents="47"/>
      <j2 rule="rewrite" parents="86"/>
    </justification>
  </clause>

  <clause id="88">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(77(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="77 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x ^ (y ^ x') = 0
    ]]></literal>
    <justification jstring="[back_rewrite(72),rewrite([88(5)])].">
      <j1 rule="back_rewrite" parents="72"/>
      <j2 rule="rewrite" parents="88"/>
    </justification>
  </clause>

  <clause id="100">
    <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="110">
    <literal><![CDATA[
      x ^ (y v (z ^ (x v (y' v u)))) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(75(a,1),12(a,1,2,2,2,2)),rewrite([30(3),2(2),24(2)]),flip(a)].">
      <j1 rule="para" parents="75 12"/>
      <j2 rule="rewrite" parents="30 2 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),20(a,1,2)),rewrite([88(2)]),flip(a)].">
      <j1 rule="para" parents="9 20"/>
      <j2 rule="rewrite" parents="88"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      x ^ ((x v y)' ^ z) = 0
    ]]></literal>
    <justification jstring="[para(79(a,1),20(a,1,2)),rewrite([88(2)]),flip(a)].">
      <j1 rule="para" parents="79 20"/>
      <j2 rule="rewrite" parents="88"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="129">
    <literal><![CDATA[
      x ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),113(a,1,2,1))].">
      <j1 rule="para" parents="2 113"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      x ^ (y ^ ((x ^ y) v z)') = 0
    ]]></literal>
    <justification jstring="[para(113(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="113 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="132">
    <literal><![CDATA[
      x ^ (y v ((x v z)' ^ (y v u))) = x ^ (y v ((x v z)' ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(113(a,1),12(a,1,2,2,1)),rewrite([64(6)])].">
      <j1 rule="para" parents="113 12"/>
      <j2 rule="rewrite" parents="64"/>
    </justification>
  </clause>

  <clause id="134">
    <literal><![CDATA[
      x ^ (y ^ ((y ^ x) v z)) = x ^ y
    ]]></literal>
    <justification jstring="[para(4(a,1),21(a,1,2,2,1))].">
      <j1 rule="para" parents="4 21"/>
    </justification>
  </clause>

  <clause id="143">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))') = 0
    ]]></literal>
    <justification jstring="[para(129(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="129 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="147">
    <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="149">
    <literal><![CDATA[
      x ^ (y v (x v y)') != 0
    ]]></literal>
    <literal><![CDATA[
      y v (x v y)' = x'
    ]]></literal>
    <justification jstring="[para(29(a,1),10(a,1)),flip(c),xx(a)].">
      <j1 rule="para" parents="29 10"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),22(a,1,2)),rewrite([86(2)]),flip(a)].">
      <j1 rule="para" parents="8 22"/>
      <j2 rule="rewrite" parents="86"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="171">
    <literal><![CDATA[
      (x v y) ^ ((x ^ z) v y) = (x ^ z) v y
    ]]></literal>
    <justification jstring="[para(22(a,1),18(a,1,2)),rewrite([4(4)])].">
      <j1 rule="para" parents="22 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="174">
    <literal><![CDATA[
      ((x ^ y) v z) ^ (x v z)' = 0
    ]]></literal>
    <justification jstring="[para(22(a,1),129(a,1,2,1))].">
      <j1 rule="para" parents="22 129"/>
    </justification>
  </clause>

  <clause id="177">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),166(a,1,2,1))].">
      <j1 rule="para" parents="4 166"/>
    </justification>
  </clause>

  <clause id="195">
    <literal><![CDATA[
      x ^ (y ^ (z ^ x)) = y ^ (z ^ x)
    ]]></literal>
    <justification jstring="[para(5(a,1),53(a,1,2)),rewrite([5(5)])].">
      <j1 rule="para" parents="5 53"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="204">
    <literal><![CDATA[
      (x v y) ^ (y v ((x v y) ^ z)) = y v ((x v y) ^ z)
    ]]></literal>
    <justification jstring="[para(23(a,1),18(a,1,2)),rewrite([4(5)])].">
      <j1 rule="para" parents="23 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="207">
    <literal><![CDATA[
      x v y != 1
    ]]></literal>
    <literal><![CDATA[
      0 != x
    ]]></literal>
    <literal><![CDATA[
      x' = x v y
    ]]></literal>
    <justification jstring="[para(56(a,1),10(a,1)),rewrite([6(5)]),flip(b)].">
      <j1 rule="para" parents="56 10"/>
      <j2 rule="rewrite" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="259">
    <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="264">
    <literal><![CDATA[
      x ^ (y v (x v y)') != 0
    ]]></literal>
    <literal><![CDATA[
      (y v (x v y)')' = x
    ]]></literal>
    <justification jstring="[para(29(a,1),34(a,1)),rewrite([4(7)]),xx(a)].">
      <j1 rule="para" parents="29 34"/>
      <j2 rule="rewrite" parents="4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="275">
    <literal><![CDATA[
      x' v (y v x) = 1
    ]]></literal>
    <justification jstring="[para(259(a,1),87(a,1,2,2))].">
      <j1 rule="para" parents="259 87"/>
    </justification>
  </clause>

  <clause id="276">
    <literal><![CDATA[
      x' ^ (y ^ x) = 0
    ]]></literal>
    <justification jstring="[para(259(a,1),89(a,1,2,2))].">
      <j1 rule="para" parents="259 89"/>
    </justification>
  </clause>

  <clause id="280">
    <literal><![CDATA[
      x' ^ (y v x) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(275(a,1),10(a,1)),rewrite([259(10)]),flip(c),xx(a)].">
      <j1 rule="para" parents="275 10"/>
      <j2 rule="rewrite" parents="259"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="297">
    <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="395">
    <literal><![CDATA[
      x v (y v (y v (x ^ z))') = 1
    ]]></literal>
    <justification jstring="[para(147(a,1),22(a,1,2)),rewrite([86(2)]),flip(a)].">
      <j1 rule="para" parents="147 22"/>
      <j2 rule="rewrite" parents="86"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="522">
    <literal><![CDATA[
      x ^ ((x ^ y) v (z ^ y)) = y ^ x
    ]]></literal>
    <justification jstring="[para(64(a,1),40(a,1,2,2,2,1)),rewrite([64(5),64(8),70(7)])].">
      <j1 rule="para" parents="64 40"/>
      <j2 rule="rewrite" parents="64 64 70"/>
    </justification>
  </clause>

  <clause id="589">
    <literal><![CDATA[
      x' ^ (x v (y ^ x')) = x' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(27(a,1),43(a,1,2,2,2))].">
      <j1 rule="para" parents="27 43"/>
    </justification>
  </clause>

  <clause id="776">
    <literal><![CDATA[
      ((x v y) ^ z) v (y ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(81(a,1),177(a,1,2,1))].">
      <j1 rule="para" parents="81 177"/>
    </justification>
  </clause>

  <clause id="777">
    <literal><![CDATA[
      (x v y) ^ (z ^ y) = z ^ y
    ]]></literal>
    <justification jstring="[para(53(a,1),81(a,2)),rewrite([195(4)])].">
      <j1 rule="para" parents="53 81"/>
      <j2 rule="rewrite" parents="195"/>
    </justification>
  </clause>

  <clause id="797">
    <literal><![CDATA[
      (x ^ (y v z)) v (x ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(85(a,1),177(a,1,2,1))].">
      <j1 rule="para" parents="85 177"/>
    </justification>
  </clause>

  <clause id="821">
    <literal><![CDATA[
      ((x ^ y) v z) ^ (y v z)' = 0
    ]]></literal>
    <justification jstring="[para(100(a,1),129(a,1,2,1))].">
      <j1 rule="para" parents="100 129"/>
    </justification>
  </clause>

  <clause id="986">
    <literal><![CDATA[
      x ^ (y ^ ((y ^ x) v z)') = 0
    ]]></literal>
    <justification jstring="[para(4(a,1),131(a,1,2,2,1,1))].">
      <j1 rule="para" parents="4 131"/>
    </justification>
  </clause>

  <clause id="1002">
    <literal><![CDATA[
      x ^ (x' v (y v (z ^ x))) = x ^ (x' v (y v z))
    ]]></literal>
    <justification jstring="[para(75(a,1),66(a,1,2,2,2)),rewrite([30(4),3(3),3(8)]),flip(a)].">
      <j1 rule="para" parents="75 66"/>
      <j2 rule="rewrite" parents="30 3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1041">
    <literal><![CDATA[
      x ^ (y ^ (z v (y ^ x))') = 0
    ]]></literal>
    <justification jstring="[para(4(a,1),143(a,1,2,2,1,2))].">
      <j1 rule="para" parents="4 143"/>
    </justification>
  </clause>

  <clause id="1191">
    <literal><![CDATA[
      (x v (y ^ z)) ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),174(a,1,1))].">
      <j1 rule="para" parents="2 174"/>
    </justification>
  </clause>

  <clause id="2979">
    <literal><![CDATA[
      ((x v y) ^ (z v y)) v y' = 1
    ]]></literal>
    <justification jstring="[para(18(a,1),776(a,1,2,1))].">
      <j1 rule="para" parents="18 776"/>
    </justification>
  </clause>

  <clause id="3154">
    <literal><![CDATA[
      ((x ^ y) v (z ^ y)) ^ y' = 0
    ]]></literal>
    <justification jstring="[para(24(a,1),821(a,1,2,1))].">
      <j1 rule="para" parents="24 821"/>
    </justification>
  </clause>

  <clause id="3205">
    <literal><![CDATA[
      x' ^ (x v (y ^ (x' v z))) = x' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(56(a,1),110(a,1,2,2,2))].">
      <j1 rule="para" parents="56 110"/>
    </justification>
  </clause>

  <clause id="4971">
    <literal><![CDATA[
      (x ^ y) v (y ^ ((y ^ x) v z)) = y ^ ((y ^ x) v z)
    ]]></literal>
    <justification jstring="[para(134(a,1),24(a,1,2)),rewrite([2(5)])].">
      <j1 rule="para" parents="134 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="7145">
    <literal><![CDATA[
      x' ^ (x v y) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(2(a,1),280(a,1,2))].">
      <j1 rule="para" parents="2 280"/>
    </justification>
  </clause>

  <clause id="8048">
    <literal><![CDATA[
      x ^ ((x ^ y) v (x ^ z)) = (x ^ y) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(7(a,1),171(a,1,1))].">
      <j1 rule="para" parents="7 171"/>
    </justification>
  </clause>

  <clause id="10006">
    <literal><![CDATA[
      x ^ ((y ^ x) v (x ^ z)) = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(24(a,1),204(a,1,1)),rewrite([24(3),24(7)])].">
      <j1 rule="para" parents="24 204"/>
      <j2 rule="rewrite" parents="24 24"/>
    </justification>
  </clause>

  <clause id="11961">
    <literal><![CDATA[
      x' ^ ((x v y) ^ (x v (y ^ x'))') = 0
    ]]></literal>
    <justification jstring="[para(589(a,1),276(a,1,2)),rewrite([4(8),5(8)])].">
      <j1 rule="para" parents="589 276"/>
      <j2 rule="rewrite" parents="4 5"/>
    </justification>
  </clause>

  <clause id="12014">
    <literal><![CDATA[
      x' ^ (x v (x' v y)') = 0
    ]]></literal>
    <justification jstring="[para(132(a,1),589(a,2)),rewrite([4(8),120(8),33(3),4(2),9(2),3205(10)]),flip(a)].">
      <j1 rule="para" parents="132 589"/>
      <j2 rule="rewrite" parents="4 120 33 4 9 3205"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="12073">
    <literal><![CDATA[
      x v (x' v y)' = x
    ]]></literal>
    <justification jstring="[hyper(7145,a,12014,a),rewrite([2(4)])].">
      <j1 rule="hyper" parents="7145 12014"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="12161">
    <literal><![CDATA[
      x v (y v x')' = x
    ]]></literal>
    <justification jstring="[para(2(a,1),12073(a,1,2,1))].">
      <j1 rule="para" parents="2 12073"/>
    </justification>
  </clause>

  <clause id="12168">
    <literal><![CDATA[
      x ^ (x' v y)' = (x' v y)'
    ]]></literal>
    <justification jstring="[para(12073(a,1),18(a,1,2)),rewrite([4(4)])].">
      <j1 rule="para" parents="12073 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="12172">
    <literal><![CDATA[
      x' v (x v y)' = x'
    ]]></literal>
    <justification jstring="[para(259(a,1),12073(a,1,2,1,1))].">
      <j1 rule="para" parents="259 12073"/>
    </justification>
  </clause>

  <clause id="12204">
    <literal><![CDATA[
      (x ^ y) v (x ^ (y' v z)')' = 1
    ]]></literal>
    <justification jstring="[para(12073(a,1),797(a,1,1,2))].">
      <j1 rule="para" parents="12073 797"/>
    </justification>
  </clause>

  <clause id="12206">
    <literal><![CDATA[
      x' ^ ((y ^ x) v (x' v z)') = 0
    ]]></literal>
    <justification jstring="[para(12073(a,1),821(a,1,2,1)),rewrite([4(7)])].">
      <j1 rule="para" parents="12073 821"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="12261">
    <literal><![CDATA[
      x ^ (y v x')' = (y v x')'
    ]]></literal>
    <justification jstring="[para(12161(a,1),18(a,1,2)),rewrite([4(4)])].">
      <j1 rule="para" parents="12161 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="12265">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(259(a,1),12161(a,1,2,1,2))].">
      <j1 rule="para" parents="259 12161"/>
    </justification>
  </clause>

  <clause id="12392">
    <literal><![CDATA[
      x' ^ (y ^ (x v z)') = y ^ (x v z)'
    ]]></literal>
    <justification jstring="[para(12172(a,1),777(a,1,1))].">
      <j1 rule="para" parents="12172 777"/>
    </justification>
  </clause>

  <clause id="12396">
    <literal><![CDATA[
      x ^ ((y ^ x') v (x v z)') = 0
    ]]></literal>
    <justification jstring="[para(12172(a,1),821(a,1,2,1)),rewrite([259(7),4(6)])].">
      <j1 rule="para" parents="12172 821"/>
      <j2 rule="rewrite" parents="259 4"/>
    </justification>
  </clause>

  <clause id="12450">
    <literal><![CDATA[
      (x v y) ^ (x v (y ^ x'))' = 0
    ]]></literal>
    <justification jstring="[back_rewrite(11961),rewrite([12392(8)])].">
      <j1 rule="back_rewrite" parents="11961"/>
      <j2 rule="rewrite" parents="12392"/>
    </justification>
  </clause>

  <clause id="12453">
    <literal><![CDATA[
      x' v (x ^ y)' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(7(a,1),12265(a,1,2,1)),rewrite([2(4)])].">
      <j1 rule="para" parents="7 12265"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="12458">
    <literal><![CDATA[
      x' ^ (y v x)' = (y v x)'
    ]]></literal>
    <justification jstring="[para(12265(a,1),18(a,1,2)),rewrite([4(4)])].">
      <j1 rule="para" parents="12265 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="12744">
    <literal><![CDATA[
      x ^ (y' v (y ^ x))' = 0
    ]]></literal>
    <justification jstring="[para(12168(a,1),1041(a,1,2))].">
      <j1 rule="para" parents="12168 1041"/>
    </justification>
  </clause>

  <clause id="13479">
    <literal><![CDATA[
      x ^ ((y ^ x) v y')' = 0
    ]]></literal>
    <justification jstring="[para(12261(a,1),986(a,1,2))].">
      <j1 rule="para" parents="12261 986"/>
    </justification>
  </clause>

  <clause id="14769">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(12453(a,1),6(a,1,2))].">
      <j1 rule="para" parents="12453 6"/>
    </justification>
  </clause>

  <clause id="14774">
    <literal><![CDATA[
      x v (x' ^ y)' = (x' ^ y)'
    ]]></literal>
    <justification jstring="[para(259(a,1),12453(a,1,1))].">
      <j1 rule="para" parents="259 12453"/>
    </justification>
  </clause>

  <clause id="14873">
    <literal><![CDATA[
      x' ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(4(a,1),14769(a,1,2,1))].">
      <j1 rule="para" parents="4 14769"/>
    </justification>
  </clause>

  <clause id="14921">
    <literal><![CDATA[
      x ^ (y ^ x')' = x
    ]]></literal>
    <justification jstring="[para(259(a,1),14873(a,1,1)),rewrite([259(6)])].">
      <j1 rule="para" parents="259 14873"/>
      <j2 rule="rewrite" parents="259"/>
    </justification>
  </clause>

  <clause id="15077">
    <literal><![CDATA[
      x v (y ^ x')' = (y ^ x')'
    ]]></literal>
    <justification jstring="[para(14921(a,1),24(a,1,2)),rewrite([2(4)])].">
      <j1 rule="para" parents="14921 24"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="16554">
    <literal><![CDATA[
      x' ^ (y ^ ((x' ^ y)' v (x ^ z))) = 0
    ]]></literal>
    <justification jstring="[para(14774(a,1),1191(a,1,2,1)),rewrite([259(9),4(8),5(8)])].">
      <j1 rule="para" parents="14774 1191"/>
      <j2 rule="rewrite" parents="259 4 5"/>
    </justification>
  </clause>

  <clause id="17437">
    <literal><![CDATA[
      x ^ (y' ^ ((x ^ y')' v (y ^ z))) = 0
    ]]></literal>
    <justification jstring="[para(15077(a,1),1191(a,1,2,1)),rewrite([259(9),4(8),5(8)])].">
      <j1 rule="para" parents="15077 1191"/>
      <j2 rule="rewrite" parents="259 4 5"/>
    </justification>
  </clause>

  <clause id="18950">
    <literal><![CDATA[
      (x ^ y) v (y' v x') = 1
    ]]></literal>
    <justification jstring="[para(12261(a,1),12204(a,1,2,1)),rewrite([259(6)])].">
      <j1 rule="para" parents="12261 12204"/>
      <j2 rule="rewrite" parents="259"/>
    </justification>
  </clause>

  <clause id="18951">
    <literal><![CDATA[
      (x' ^ y) v (y' v x) = 1
    ]]></literal>
    <justification jstring="[para(12458(a,1),12204(a,1,2,1)),rewrite([259(6)])].">
      <j1 rule="para" parents="12458 12204"/>
      <j2 rule="rewrite" parents="259"/>
    </justification>
  </clause>

  <clause id="18999">
    <literal><![CDATA[
      x' v ((x ^ y) v y') = 1
    ]]></literal>
    <justification jstring="[hyper(207,a,18950,a,b,12744,a(flip)),rewrite([12744(5),76(2),12744(6),259(7),3(7),64(8)]),flip(a)].">
      <j1 rule="hyper" parents="207 18950 12744"/>
      <j2 rule="rewrite" parents="12744 76 12744 259 3 64"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="19177">
    <literal><![CDATA[
      (x ^ y') v (x' v y) = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),18951(a,1,1))].">
      <j1 rule="para" parents="4 18951"/>
    </justification>
  </clause>

  <clause id="19580">
    <literal><![CDATA[
      x ^ (y' ^ (x' v y)) != 0
    ]]></literal>
    <literal><![CDATA[
      (x' v y)' = x ^ y'
    ]]></literal>
    <justification jstring="[para(19177(a,1),264(a,1,2,2,1)),rewrite([73(6),2(6),64(6),5(5),19177(14),73(11),2(11),64(11)])].">
      <j1 rule="para" parents="19177 264"/>
      <j2 rule="rewrite" parents="73 2 64 5 19177 73 2 64"/>
    </justification>
  </clause>

  <clause id="19627">
    <literal><![CDATA[
      (x ^ y) v (y' v (x ^ y))' = y
    ]]></literal>
    <justification jstring="[hyper(149,a,12206,a),rewrite([259(8)])].">
      <j1 rule="hyper" parents="149 12206"/>
      <j2 rule="rewrite" parents="259"/>
    </justification>
  </clause>

  <clause id="21956">
    <literal><![CDATA[
      (x ^ y') v (y v (x ^ y'))' = y'
    ]]></literal>
    <justification jstring="[hyper(149,a,12396,a)].">
      <j1 rule="hyper" parents="149 12396"/>
    </justification>
  </clause>

  <clause id="22393">
    <literal><![CDATA[
      (x v (y ^ x'))' = (x v y)'
    ]]></literal>
    <justification jstring="[hyper(297,a,395,a,b,12450,a),flip(a)].">
      <j1 rule="hyper" parents="297 395 12450"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22408">
    <literal><![CDATA[
      (x ^ y') v (y v x)' = y'
    ]]></literal>
    <justification jstring="[back_rewrite(21956),rewrite([22393(6)])].">
      <j1 rule="back_rewrite" parents="21956"/>
      <j2 rule="rewrite" parents="22393"/>
    </justification>
  </clause>

  <clause id="22418">
    <literal><![CDATA[
      x v (y ^ x') = x v y
    ]]></literal>
    <justification jstring="[para(22393(a,1),259(a,1,1)),rewrite([259(3)]),flip(a)].">
      <j1 rule="para" parents="22393 259"/>
      <j2 rule="rewrite" parents="259"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22419">
    <literal><![CDATA[
      (x' v (y ^ x))' = (x' v y)'
    ]]></literal>
    <justification jstring="[para(259(a,1),22393(a,1,1,2,2))].">
      <j1 rule="para" parents="259 22393"/>
    </justification>
  </clause>

  <clause id="22442">
    <literal><![CDATA[
      ((x ^ y) v x')' = (y v x')'
    ]]></literal>
    <justification jstring="[para(13479(a,1),22393(a,1,1,2)),rewrite([2(5),64(5),2(8),100(8)])].">
      <j1 rule="para" parents="13479 22393"/>
      <j2 rule="rewrite" parents="2 64 2 100"/>
    </justification>
  </clause>

  <clause id="22463">
    <literal><![CDATA[
      (x ^ y) v (y' v x)' = y
    ]]></literal>
    <justification jstring="[back_rewrite(19627),rewrite([22419(5)])].">
      <j1 rule="back_rewrite" parents="19627"/>
      <j2 rule="rewrite" parents="22419"/>
    </justification>
  </clause>

  <clause id="22489">
    <literal><![CDATA[
      x v (y v x)' = x v y'
    ]]></literal>
    <justification jstring="[para(22408(a,1),22(a,1,2)),flip(a)].">
      <j1 rule="para" parents="22408 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22566">
    <literal><![CDATA[
      (x ^ y)' = x' v y'
    ]]></literal>
    <justification jstring="[para(14769(a,1),22408(a,1,1)),rewrite([22442(5),22489(5)]),flip(a)].">
      <j1 rule="para" parents="14769 22408"/>
      <j2 rule="rewrite" parents="22442 22489"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22575">
    <literal><![CDATA[
      x ^ ((x ^ y) v y') = x
    ]]></literal>
    <justification jstring="[para(18999(a,1),22408(a,1,2,1)),rewrite([259(5),4(4),73(6),2(6),64(6),259(6)])].">
      <j1 rule="para" parents="18999 22408"/>
      <j2 rule="rewrite" parents="259 4 73 2 64 259"/>
    </justification>
  </clause>

  <clause id="22636">
    <literal><![CDATA[
      x ^ (y' ^ (x' v y)) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(17437),rewrite([22566(4),259(4),3(5),7(4)])].">
      <j1 rule="back_rewrite" parents="17437"/>
      <j2 rule="rewrite" parents="22566 259 3 7"/>
    </justification>
  </clause>

  <clause id="23745">
    <literal><![CDATA[
      x' ^ (y ^ (y' v x)) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(16554),rewrite([22566(4),259(3),3(5),46(5)])].">
      <j1 rule="back_rewrite" parents="16554"/>
      <j2 rule="rewrite" parents="22566 259 3 46"/>
    </justification>
  </clause>

  <clause id="27099">
    <literal><![CDATA[
      (x' v y)' = x ^ y'
    ]]></literal>
    <justification jstring="[back_rewrite(19580),rewrite([22636(5)]),xx(a)].">
      <j1 rule="back_rewrite" parents="19580"/>
      <j2 rule="rewrite" parents="22636"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="27281">
    <literal><![CDATA[
      (x ^ y) v (y ^ x') = y
    ]]></literal>
    <justification jstring="[back_rewrite(22463),rewrite([27099(4)])].">
      <j1 rule="back_rewrite" parents="22463"/>
      <j2 rule="rewrite" parents="27099"/>
    </justification>
  </clause>

  <clause id="27749">
    <literal><![CDATA[
      (x v y)' = x' ^ y'
    ]]></literal>
    <justification jstring="[para(259(a,1),27099(a,1,1,1))].">
      <j1 rule="para" parents="259 27099"/>
    </justification>
  </clause>

  <clause id="33655">
    <literal><![CDATA[
      x ^ ((y ^ x) v (z ^ x)) = (y ^ x) v (z ^ x)
    ]]></literal>
    <justification jstring="[para(3154(a,1),22575(a,1,2,1)),rewrite([259(6),64(5),4(4)])].">
      <j1 rule="para" parents="3154 22575"/>
      <j2 rule="rewrite" parents="259 64 4"/>
    </justification>
  </clause>

  <clause id="33837">
    <literal><![CDATA[
      (x v y) ^ (y v x') = y
    ]]></literal>
    <justification jstring="[hyper(34,a,2979,a,b,23745,a),rewrite([259(2),27749(3),2(5),22418(5)]),flip(a)].">
      <j1 rule="hyper" parents="34 2979 23745"/>
      <j2 rule="rewrite" parents="259 27749 2 22418"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34034">
    <literal><![CDATA[
      (x ^ y) v (x ^ ((y v z) ^ y')) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(70(a,1),27281(a,1,1)),rewrite([5(5)])].">
      <j1 rule="para" parents="70 27281"/>
      <j2 rule="rewrite" parents="5"/>
    </justification>
  </clause>

  <clause id="34041">
    <literal><![CDATA[
      (x ^ y) v (((y ^ x) v (x ^ z)) ^ y') = (y ^ x) v (x ^ z)
    ]]></literal>
    <justification jstring="[para(78(a,1),27281(a,1,1))].">
      <j1 rule="para" parents="78 27281"/>
    </justification>
  </clause>

  <clause id="34068">
    <literal><![CDATA[
      (x ^ y) v (((y ^ x) v (z ^ x)) ^ y') = (y ^ x) v (z ^ x)
    ]]></literal>
    <justification jstring="[para(522(a,1),27281(a,1,1))].">
      <j1 rule="para" parents="522 27281"/>
    </justification>
  </clause>

  <clause id="34078">
    <literal><![CDATA[
      x ^ (x' v y) = x ^ y
    ]]></literal>
    <justification jstring="[para(23745(a,1),27281(a,1,1)),rewrite([259(6),4(5),85(5),64(3)]),flip(a)].">
      <j1 rule="para" parents="23745 27281"/>
      <j2 rule="rewrite" parents="259 4 85 64"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34138">
    <literal><![CDATA[
      x ^ (y v (z ^ x)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(1002),rewrite([34078(5),34078(7)])].">
      <j1 rule="back_rewrite" parents="1002"/>
      <j2 rule="rewrite" parents="34078 34078"/>
    </justification>
  </clause>

  <clause id="34321">
    <literal><![CDATA[
      (x ^ y) v (z ^ y) = y ^ ((x ^ y) v z)
    ]]></literal>
    <justification jstring="[back_rewrite(33655),rewrite([34138(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="33655"/>
      <j2 rule="rewrite" parents="34138"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="34549">
    <literal><![CDATA[
      (x ^ y) v (x ^ (((y ^ x) v z) ^ y')) = x ^ ((y ^ x) v z)
    ]]></literal>
    <justification jstring="[back_rewrite(34068),rewrite([34321(4),5(6),34321(10)])].">
      <j1 rule="back_rewrite" parents="34068"/>
      <j2 rule="rewrite" parents="34321 5 34321"/>
    </justification>
  </clause>

  <clause id="35979">
    <literal><![CDATA[
      (x v y) ^ x' = y ^ x'
    ]]></literal>
    <justification jstring="[para(22418(a,1),33837(a,1,1)),rewrite([2(5),24(5)])].">
      <j1 rule="para" parents="22418 33837"/>
      <j2 rule="rewrite" parents="2 24"/>
    </justification>
  </clause>

  <clause id="35996">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z ^ y')) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(34034),rewrite([35979(4)])].">
      <j1 rule="back_rewrite" parents="34034"/>
      <j2 rule="rewrite" parents="35979"/>
    </justification>
  </clause>

  <clause id="36110">
    <literal><![CDATA[
      x ^ ((y ^ x) v z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(34549),rewrite([35996(7),22(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="34549"/>
      <j2 rule="rewrite" parents="35996 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36217">
    <literal><![CDATA[
      (x ^ y) v (y ^ z) = y ^ (x v (y ^ z))
    ]]></literal>
    <justification jstring="[back_rewrite(10006),rewrite([36110(4)]),flip(a)].">
      <j1 rule="back_rewrite" parents="10006"/>
      <j2 rule="rewrite" parents="36110"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36259">
    <literal><![CDATA[
      x ^ (y v (x ^ z)) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(34041),rewrite([36217(4),5(6),35979(5),5(4),51(5),35996(5),36217(5)]),flip(a)].">
      <j1 rule="back_rewrite" parents="34041"/>
      <j2 rule="rewrite" parents="36217 5 35979 5 51 35996 36217"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36272">
    <literal><![CDATA[
      x ^ ((x ^ y) v z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(4971),rewrite([36217(5),36259(5),100(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="4971"/>
      <j2 rule="rewrite" parents="36217 36259 100"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36306">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[back_rewrite(8048),rewrite([36259(4),36272(3)]),flip(a)].">
      <j1 rule="back_rewrite" parents="8048"/>
      <j2 rule="rewrite" parents="36259 36272"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36307">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[resolve(36306,a,13,a)].">
      <j1 rule="resolve" parents="36306 13"/>
    </justification>
  </clause>

</proof>

</proofs>
