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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 August-2006A, August 2006.
Process 11505 was started by mccune on cleo.thornwood,
Sat Aug 12 21:02:47 2006
The command was "/home/mccune/bin/prover9 -f lt.in uc.in H27d.in".
]]></heading>

<proof number="1" length="97" max_count="25">

<comments><![CDATA[
% Proof 1 at 13.15 (+ 0.15) seconds: distributivity.
% Length of proof is 97.
% Level of proof is 24.
% Maximum clause weight is 23.
% Given clauses 695.
]]></comments>

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

  <clause id="2">
    <literal><![CDATA[
      x v y = y v x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

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

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

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

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

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

  <clause id="10">
    <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">
    <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">
    <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="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="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="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="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="42">
    <literal><![CDATA[
      x ^ (y v ((z ^ x) v (z ^ (y v u)))) = x ^ (y v (z ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(4(a,1),12(a,1,2,2,1))].">
      <j1 rule="para" parents="4 12"/>
    </justification>
  </clause>

  <clause id="46">
    <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="54">
    <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="61">
    <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="72">
    <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="77">
    <literal><![CDATA[
      1 v x = 1
    ]]></literal>
    <justification jstring="[para(61(a,1),6(a,1))].">
      <j1 rule="para" parents="61 6"/>
    </justification>
  </clause>

  <clause id="78">
    <literal><![CDATA[
      x v (x' v y) = 1
    ]]></literal>
    <justification jstring="[back_rewrite(28),rewrite(77(5))].">
      <j1 rule="back_rewrite" parents="28"/>
      <j2 rule="rewrite" parents="77"/>
    </justification>
  </clause>

  <clause id="80">
    <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="82">
    <literal><![CDATA[
      x ^ (x' ^ y) = 0
    ]]></literal>
    <justification jstring="[back_rewrite(31),rewrite(80(5))].">
      <j1 rule="back_rewrite" parents="31"/>
      <j2 rule="rewrite" parents="80"/>
    </justification>
  </clause>

  <clause id="83">
    <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="85">
    <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="88">
    <literal><![CDATA[
      x v 1 = 1
    ]]></literal>
    <justification jstring="[para(77(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="77 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="89">
    <literal><![CDATA[
      x v (y v x') = 1
    ]]></literal>
    <justification jstring="[back_rewrite(72),rewrite(88(5))].">
      <j1 rule="back_rewrite" parents="72"/>
      <j2 rule="rewrite" parents="88"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      x ^ 0 = 0
    ]]></literal>
    <justification jstring="[para(80(a,1),4(a,1)),flip(a)].">
      <j1 rule="para" parents="80 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="92">
    <literal><![CDATA[
      x ^ (y ^ x') = 0
    ]]></literal>
    <justification jstring="[back_rewrite(85),rewrite(91(5))].">
      <j1 rule="back_rewrite" parents="85"/>
      <j2 rule="rewrite" parents="91"/>
    </justification>
  </clause>

  <clause id="95">
    <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="101">
    <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(78(a,1),12(a,1,2,2,2,2)),rewrite(30(3),2(2),24(2)),flip(a)].">
      <j1 rule="para" parents="78 12"/>
      <j2 rule="rewrite" parents="30 2 24"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="142">
    <literal><![CDATA[
      x ^ (x v y)' = 0
    ]]></literal>
    <justification jstring="[para(9(a,1),20(a,1,2)),rewrite(91(2)),flip(a)].">
      <j1 rule="para" parents="9 20"/>
      <j2 rule="rewrite" parents="91"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="147">
    <literal><![CDATA[
      x ^ ((x v y)' ^ z) = 0
    ]]></literal>
    <justification jstring="[para(82(a,1),20(a,1,2)),rewrite(91(2)),flip(a)].">
      <j1 rule="para" parents="82 20"/>
      <j2 rule="rewrite" parents="91"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="149">
    <literal><![CDATA[
      x ^ (y v x)' = 0
    ]]></literal>
    <justification jstring="[para(2(a,1),142(a,1,2,1))].">
      <j1 rule="para" parents="2 142"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      x ^ (y v ((x v z)' ^ (y v u))) = x ^ (y v ((x v z)' ^ (x v u)))
    ]]></literal>
    <justification jstring="[para(142(a,1),12(a,1,2,2,1)),rewrite(64(6))].">
      <j1 rule="para" parents="142 12"/>
      <j2 rule="rewrite" parents="64"/>
    </justification>
  </clause>

  <clause id="157">
    <literal><![CDATA[
      x ^ (y ^ (z v (x ^ y))') = 0
    ]]></literal>
    <justification jstring="[para(149(a,1),5(a,1)),flip(a)].">
      <j1 rule="para" parents="149 5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="164">
    <literal><![CDATA[
      x v (y ^ (x ^ y)') != 1
    ]]></literal>
    <literal><![CDATA[
      y ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(32(a,1),10(b,1)),flip(c),xx(b)].">
      <j1 rule="para" parents="32 10"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="190">
    <literal><![CDATA[
      x v (x ^ y)' = 1
    ]]></literal>
    <justification jstring="[para(8(a,1),22(a,1,2)),rewrite(88(2)),flip(a)].">
      <j1 rule="para" parents="8 22"/>
      <j2 rule="rewrite" parents="88"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="198">
    <literal><![CDATA[
      x v (y v ((x ^ z) v y)') = 1
    ]]></literal>
    <justification jstring="[para(29(a,1),22(a,1,2)),rewrite(88(2)),flip(a)].">
      <j1 rule="para" parents="29 22"/>
      <j2 rule="rewrite" parents="88"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="202">
    <literal><![CDATA[
      x v (y ^ x)' = 1
    ]]></literal>
    <justification jstring="[para(4(a,1),190(a,1,2,1))].">
      <j1 rule="para" parents="4 190"/>
    </justification>
  </clause>

  <clause id="208">
    <literal><![CDATA[
      x v (y v (z ^ (x v y))') = 1
    ]]></literal>
    <justification jstring="[para(202(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="202 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="215">
    <literal><![CDATA[
      ((x v y) ^ z) v (x ^ z)' = 1
    ]]></literal>
    <justification jstring="[para(20(a,1),202(a,1,2,1))].">
      <j1 rule="para" parents="20 202"/>
    </justification>
  </clause>

  <clause id="298">
    <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="301">
    <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="320">
    <literal><![CDATA[
      x' v (y v x) = 1
    ]]></literal>
    <justification jstring="[para(301(a,1),89(a,1,2,2))].">
      <j1 rule="para" parents="301 89"/>
    </justification>
  </clause>

  <clause id="321">
    <literal><![CDATA[
      x' ^ (y ^ x) = 0
    ]]></literal>
    <justification jstring="[para(301(a,1),92(a,1,2,2))].">
      <j1 rule="para" parents="301 92"/>
    </justification>
  </clause>

  <clause id="326">
    <literal><![CDATA[
      x' ^ (y v x) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(320(a,1),10(a,1)),rewrite(301(10)),flip(c),xx(a)].">
      <j1 rule="para" parents="320 10"/>
      <j2 rule="rewrite" parents="301"/>
      <j2 rule="flip"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="335">
    <literal><![CDATA[
      x ^ (y ^ (y ^ ((x ^ y) v z))') = 0
    ]]></literal>
    <justification jstring="[para(21(a,1),321(a,1,2)),rewrite(4(6),5(6))].">
      <j1 rule="para" parents="21 321"/>
      <j2 rule="rewrite" parents="4 5"/>
    </justification>
  </clause>

  <clause id="733">
    <literal><![CDATA[
      x ^ ((y ^ x) v (y ^ z)) = y ^ x
    ]]></literal>
    <justification jstring="[para(64(a,1),42(a,1,2,2,2,2)),rewrite(64(5),64(8),83(7))].">
      <j1 rule="para" parents="64 42"/>
      <j2 rule="rewrite" parents="64 64 83"/>
    </justification>
  </clause>

  <clause id="819">
    <literal><![CDATA[
      x' ^ (x v (y ^ x')) = x' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(27(a,1),46(a,1,2,2,2))].">
      <j1 rule="para" parents="27 46"/>
    </justification>
  </clause>

  <clause id="1365">
    <literal><![CDATA[
      x v ((x ^ y) v (x ^ z))' = 1
    ]]></literal>
    <justification jstring="[para(198(a,1),22(a,1)),flip(a)].">
      <j1 rule="para" parents="198 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2817">
    <literal><![CDATA[
      x' ^ (x v (y ^ (x' v z))) = x' ^ (x v y)
    ]]></literal>
    <justification jstring="[para(54(a,1),110(a,1,2,2,2))].">
      <j1 rule="para" parents="54 110"/>
    </justification>
  </clause>

  <clause id="6978">
    <literal><![CDATA[
      x' ^ (x v y) != 0
    ]]></literal>
    <literal><![CDATA[
      y v x = x
    ]]></literal>
    <justification jstring="[para(2(a,1),326(a,1,2))].">
      <j1 rule="para" parents="2 326"/>
    </justification>
  </clause>

  <clause id="10451">
    <literal><![CDATA[
      x' ^ (x v (x' v y)') = 0
    ]]></literal>
    <justification jstring="[para(153(a,1),819(a,2)),rewrite(4(8),147(8),33(3),4(2),9(2),2817(10)),flip(a)].">
      <j1 rule="para" parents="153 819"/>
      <j2 rule="rewrite" parents="4 147 33 4 9 2817"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="10465">
    <literal><![CDATA[
      x v (x' v y)' = x
    ]]></literal>
    <justification jstring="[hyper(6978,a,10451,a),rewrite(2(4))].">
      <j1 rule="hyper" parents="6978 10451"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="10475">
    <literal><![CDATA[
      x v (y v x')' = x
    ]]></literal>
    <justification jstring="[para(2(a,1),10465(a,1,2,1))].">
      <j1 rule="para" parents="2 10465"/>
    </justification>
  </clause>

  <clause id="10485">
    <literal><![CDATA[
      x' v (x v y)' = x'
    ]]></literal>
    <justification jstring="[para(301(a,1),10465(a,1,2,1,1))].">
      <j1 rule="para" parents="301 10465"/>
    </justification>
  </clause>

  <clause id="10612">
    <literal><![CDATA[
      x' v (y v x)' = x'
    ]]></literal>
    <justification jstring="[para(301(a,1),10475(a,1,2,1,2))].">
      <j1 rule="para" parents="301 10475"/>
    </justification>
  </clause>

  <clause id="10693">
    <literal><![CDATA[
      x' ^ (x v y)' = (x v y)'
    ]]></literal>
    <justification jstring="[para(10485(a,1),18(a,1,2)),rewrite(4(4))].">
      <j1 rule="para" parents="10485 18"/>
      <j2 rule="rewrite" parents="4"/>
    </justification>
  </clause>

  <clause id="10781">
    <literal><![CDATA[
      x' v (x ^ y)' = (x ^ y)'
    ]]></literal>
    <justification jstring="[para(7(a,1),10612(a,1,2,1)),rewrite(2(4))].">
      <j1 rule="para" parents="7 10612"/>
      <j2 rule="rewrite" parents="2"/>
    </justification>
  </clause>

  <clause id="11693">
    <literal><![CDATA[
      x ^ (y v (x ^ y'))' = 0
    ]]></literal>
    <justification jstring="[para(10693(a,1),157(a,1,2))].">
      <j1 rule="para" parents="10693 157"/>
    </justification>
  </clause>

  <clause id="11929">
    <literal><![CDATA[
      x' ^ (x ^ y)' = x'
    ]]></literal>
    <justification jstring="[para(10781(a,1),6(a,1,2))].">
      <j1 rule="para" parents="10781 6"/>
    </justification>
  </clause>

  <clause id="11934">
    <literal><![CDATA[
      x v (x' ^ y)' = (x' ^ y)'
    ]]></literal>
    <justification jstring="[para(301(a,1),10781(a,1,1))].">
      <j1 rule="para" parents="301 10781"/>
    </justification>
  </clause>

  <clause id="12012">
    <literal><![CDATA[
      x' ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(4(a,1),11929(a,1,2,1))].">
      <j1 rule="para" parents="4 11929"/>
    </justification>
  </clause>

  <clause id="12067">
    <literal><![CDATA[
      x v ((x' v y) ^ (z ^ x)') = 1
    ]]></literal>
    <justification jstring="[para(12012(a,1),215(a,1,2,1)),rewrite(301(7),2(6))].">
      <j1 rule="para" parents="12012 215"/>
      <j2 rule="rewrite" parents="301 2"/>
    </justification>
  </clause>

  <clause id="12922">
    <literal><![CDATA[
      (x ^ y) v (x ^ (x ^ y)') = x
    ]]></literal>
    <justification jstring="[hyper(298,a,1365,a,b,11693,a),rewrite(301(7))].">
      <j1 rule="hyper" parents="298 1365 11693"/>
      <j2 rule="rewrite" parents="301"/>
    </justification>
  </clause>

  <clause id="12979">
    <literal><![CDATA[
      x v (y' ^ (x v y))' = 1
    ]]></literal>
    <justification jstring="[para(11934(a,1),208(a,1,2))].">
      <j1 rule="para" parents="11934 208"/>
    </justification>
  </clause>

  <clause id="13646">
    <literal><![CDATA[
      (x' ^ (x v y))' = (y ^ x')'
    ]]></literal>
    <justification jstring="[hyper(37,a,12979,a,b,335,a),rewrite(2(7),819(8)),flip(a)].">
      <j1 rule="hyper" parents="37 12979 335"/>
      <j2 rule="rewrite" parents="2 819"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="13730">
    <literal><![CDATA[
      ((x v y) ^ x')' = (y ^ x')'
    ]]></literal>
    <justification jstring="[para(4(a,1),13646(a,1,1))].">
      <j1 rule="para" parents="4 13646"/>
    </justification>
  </clause>

  <clause id="13741">
    <literal><![CDATA[
      (x ^ (x' v y))' = (y ^ x)'
    ]]></literal>
    <justification jstring="[para(301(a,1),13646(a,1,1,1)),rewrite(301(6))].">
      <j1 rule="para" parents="301 13646"/>
      <j2 rule="rewrite" parents="301"/>
    </justification>
  </clause>

  <clause id="13928">
    <literal><![CDATA[
      (x v y) ^ x' = y ^ x'
    ]]></literal>
    <justification jstring="[para(13730(a,1),301(a,1,1)),rewrite(301(4)),flip(a)].">
      <j1 rule="para" parents="13730 301"/>
      <j2 rule="rewrite" parents="301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14034">
    <literal><![CDATA[
      x ^ (x' v y) = y ^ x
    ]]></literal>
    <justification jstring="[para(13741(a,1),301(a,1,1)),rewrite(301(3)),flip(a)].">
      <j1 rule="para" parents="13741 301"/>
      <j2 rule="rewrite" parents="301"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14077">
    <literal><![CDATA[
      (x' v y) ^ (y ^ x)' = x'
    ]]></literal>
    <justification jstring="[para(13741(a,1),164(a,1,2,2)),rewrite(12067(6),14034(8)),xx(a)].">
      <j1 rule="para" parents="13741 164"/>
      <j2 rule="rewrite" parents="12067 14034"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="14380">
    <literal><![CDATA[
      ((x ^ y) v z) ^ y' = z ^ y'
    ]]></literal>
    <justification jstring="[para(101(a,1),13928(a,1,1)),rewrite(13928(3)),flip(a)].">
      <j1 rule="para" parents="101 13928"/>
      <j2 rule="rewrite" parents="13928"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14735">
    <literal><![CDATA[
      x ^ (x ^ y)' = x ^ y'
    ]]></literal>
    <justification jstring="[para(14077(a,1),95(a,1,2)),flip(a)].">
      <j1 rule="para" parents="14077 95"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="16622">
    <literal><![CDATA[
      (x ^ y) v (x ^ y') = x
    ]]></literal>
    <justification jstring="[back_rewrite(12922),rewrite(14735(4))].">
      <j1 rule="back_rewrite" parents="12922"/>
      <j2 rule="rewrite" parents="14735"/>
    </justification>
  </clause>

  <clause id="20293">
    <literal><![CDATA[
      (x ^ y) v (y ^ x') = y
    ]]></literal>
    <justification jstring="[para(4(a,1),16622(a,1,1))].">
      <j1 rule="para" parents="4 16622"/>
    </justification>
  </clause>

  <clause id="21099">
    <literal><![CDATA[
      (x ^ y) v (x ^ (z ^ y')) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(83(a,1),20293(a,1,1)),rewrite(5(5),13928(4))].">
      <j1 rule="para" parents="83 20293"/>
      <j2 rule="rewrite" parents="5 13928"/>
    </justification>
  </clause>

  <clause id="21127">
    <literal><![CDATA[
      (x ^ y) v (x ^ z) = x ^ (y v z)
    ]]></literal>
    <justification jstring="[para(733(a,1),20293(a,1,1)),rewrite(14380(6),5(4),21099(5)),flip(a)].">
      <j1 rule="para" parents="733 20293"/>
      <j2 rule="rewrite" parents="14380 5 21099"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21128">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(distributivity)
    ]]></attribute>
    <justification jstring="[resolve(21127,a,13,a)].">
      <j1 rule="resolve" parents="21127 13"/>
    </justification>
  </clause>

</proof>

</proofs>
