<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>DA-cs.out</source>

<heading><![CDATA[
Prover9 (32) version August-2006A, August 2006.
Process 11353 was started by mccune on cleo.thornwood,
Sat Aug 12 20:59:51 2006
The command was "/home/mccune/bin/prover9 -f cs.in DA.in".
]]></heading>

<proof number="1" length="73" max_count="23">

<comments><![CDATA[
% Proof 1 at 0.35 (+ 0.01) seconds: A.
% Length of proof is 73.
% Level of proof is 24.
% Maximum clause weight is 23.
% Given clauses 205.
]]></comments>

  <clause id="1">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

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

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

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

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

  <clause id="6">
    <literal><![CDATA[
      x * y @ z = (x @ z) * (y @ z)
    ]]></literal>
    <justification jstring="[copy(5),flip(a)].">
      <j1 rule="copy" parents="5"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      (A @ B) @ C != D @ (F @ G)
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1)),rewrite(3(4))].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,1)),rewrite(3(2),3(5)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="15">
    <literal><![CDATA[
      x * y * z * (z @ x * y) = z * x * y
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x @ y) * (z @ y) * ((z @ x) @ y) = (z @ y) * (x @ y)
    ]]></literal>
    <justification jstring="[para(4(a,1),6(a,1,1)),rewrite(6(2),6(7)),flip(a)].">
      <j1 rule="para" parents="4 6"/>
      <j2 rule="rewrite" parents="6 6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      ((A @ B) @ C) * x != (D @ (F @ G)) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(2,b,7,a)].">
      <j1 rule="ur" parents="2 7"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      x * ((A @ B) @ C) * y != x * (D @ (F @ G)) * y
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[ur(1,b,18,a)].">
      <j1 rule="ur" parents="1 18"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="10 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="27">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="37">
    <literal><![CDATA[
      x * y * z != x * u * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ y) = z
    ]]></literal>
    <justification jstring="[para(4(a,1),8(a,1,2)),flip(a)].">
      <j1 rule="para" parents="4 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="38">
    <literal><![CDATA[
      x * y * z != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = z
    ]]></literal>
    <justification jstring="[para(4(a,1),8(a,1)),flip(a)].">
      <j1 rule="para" parents="4 8"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x * (D @ (F @ G)) * (((A @ B) @ C) @ x) != ((A @ B) @ C) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[para(4(a,1),20(a,1)),flip(a)].">
      <j1 rule="para" parents="4 20"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="58">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(8,a,27,a)].">
      <j1 rule="hyper" parents="8 27"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      x * y != y
    ]]></literal>
    <literal><![CDATA[
      z @ z = x
    ]]></literal>
    <justification jstring="[para(58(a,1),2(a,1)),flip(a)].">
      <j1 rule="para" parents="58 2"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(58(a,1),4(a,1,2)),rewrite(58(5))].">
      <j1 rule="para" parents="58 4"/>
      <j2 rule="rewrite" parents="58"/>
    </justification>
  </clause>

  <clause id="68">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) * z = z
    ]]></literal>
    <justification jstring="[para(6(a,1),58(a,1,1)),rewrite(3(6))].">
      <j1 rule="para" parents="6 58"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="73">
    <literal><![CDATA[
      x * y * z != u * x * y
    ]]></literal>
    <literal><![CDATA[
      u * (u @ x * y) = z
    ]]></literal>
    <justification jstring="[para(3(a,1),11(a,1))].">
      <j1 rule="para" parents="3 11"/>
    </justification>
  </clause>

  <clause id="83">
    <literal><![CDATA[
      (x @ x) @ y = y @ y
    ]]></literal>
    <justification jstring="[hyper(24,a,65,a),flip(a)].">
      <j1 rule="hyper" parents="24 65"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      (x @ x) @ y * z = (y @ y * z) * (z @ y * z)
    ]]></literal>
    <justification jstring="[para(83(a,2),6(a,1))].">
      <j1 rule="para" parents="83 6"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      ((x @ x) @ y) * z = z
    ]]></literal>
    <justification jstring="[para(83(a,2),58(a,1,1))].">
      <j1 rule="para" parents="83 58"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      x * (((y @ y) @ z) @ x) = x
    ]]></literal>
    <justification jstring="[para(83(a,2),65(a,1,2,1))].">
      <j1 rule="para" parents="83 65"/>
    </justification>
  </clause>

  <clause id="124">
    <literal><![CDATA[
      x * (x @ ((y @ y) @ z)) = x * ((y @ y) @ z)
    ]]></literal>
    <justification jstring="[hyper(11,a,115,a)].">
      <j1 rule="hyper" parents="11 115"/>
    </justification>
  </clause>

  <clause id="131">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      ((z @ z) @ u) @ x = y
    ]]></literal>
    <justification jstring="[para(115(a,1),11(a,2)),rewrite(115(8))].">
      <j1 rule="para" parents="115 11"/>
      <j2 rule="rewrite" parents="115"/>
    </justification>
  </clause>

  <clause id="135">
    <literal><![CDATA[
      (((x @ x) @ y) @ z) * u = u
    ]]></literal>
    <justification jstring="[para(83(a,2),115(a,1,1,1))].">
      <j1 rule="para" parents="83 115"/>
    </justification>
  </clause>

  <clause id="159">
    <literal><![CDATA[
      (x @ x) @ y = z @ z
    ]]></literal>
    <justification jstring="[hyper(64,a,115,a),flip(a)].">
      <j1 rule="hyper" parents="64 115"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="160">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(64,a,58,a)].">
      <j1 rule="hyper" parents="64 58"/>
    </justification>
  </clause>

  <clause id="163">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(160)].">
      <j1 rule="new_symbol" parents="160"/>
    </justification>
  </clause>

  <clause id="166">
    <literal><![CDATA[
      c_0 @ x = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(159),rewrite(163(1),163(3))].">
      <j1 rule="back_rewrite" parents="159"/>
      <j2 rule="rewrite" parents="163 163"/>
    </justification>
  </clause>

  <clause id="172">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_rewrite(135),rewrite(163(1),166(2),166(2))].">
      <j1 rule="back_rewrite" parents="135"/>
      <j2 rule="rewrite" parents="163 166 166"/>
    </justification>
  </clause>

  <clause id="175">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      c_0 = y
    ]]></literal>
    <justification jstring="[back_rewrite(131),rewrite(163(3),166(4),166(4))].">
      <j1 rule="back_rewrite" parents="131"/>
      <j2 rule="rewrite" parents="163 166 166"/>
    </justification>
  </clause>

  <clause id="179">
    <literal><![CDATA[
      x * (x @ c_0) = x * c_0
    ]]></literal>
    <justification jstring="[back_rewrite(124),rewrite(163(1),166(2),163(4),166(5))].">
      <j1 rule="back_rewrite" parents="124"/>
      <j2 rule="rewrite" parents="163 166 163 166"/>
    </justification>
  </clause>

  <clause id="181">
    <literal><![CDATA[
      x * c_0 = x
    ]]></literal>
    <justification jstring="[back_rewrite(116),rewrite(163(1),166(2),166(2))].">
      <j1 rule="back_rewrite" parents="116"/>
      <j2 rule="rewrite" parents="163 166 166"/>
    </justification>
  </clause>

  <clause id="183">
    <literal><![CDATA[
      (x @ x * y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(112),rewrite(163(1),166(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="112"/>
      <j2 rule="rewrite" parents="163 166"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="187">
    <literal><![CDATA[
      x * (x @ c_0) = x
    ]]></literal>
    <justification jstring="[back_rewrite(179),rewrite(181(5))].">
      <j1 rule="back_rewrite" parents="179"/>
      <j2 rule="rewrite" parents="181"/>
    </justification>
  </clause>

  <clause id="198">
    <literal><![CDATA[
      x * y * ((x @ y) @ y * x) = (x @ y) * y * x
    ]]></literal>
    <justification jstring="[para(15(a,1),14(a,1)),flip(a)].">
      <j1 rule="para" parents="15 14"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="217">
    <literal><![CDATA[
      (x @ y) * ((x @ y) @ y) = x @ y
    ]]></literal>
    <justification jstring="[para(163(a,1),17(a,1,1)),rewrite(172(6),163(6),181(7))].">
      <j1 rule="para" parents="163 17"/>
      <j2 rule="rewrite" parents="172 163 181"/>
    </justification>
  </clause>

  <clause id="243">
    <literal><![CDATA[
      x @ c_0 = c_0
    ]]></literal>
    <justification jstring="[hyper(175,a,187,a),flip(a)].">
      <j1 rule="hyper" parents="175 187"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="259">
    <literal><![CDATA[
      x * (x @ x * y) = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(37,a,15,a),flip(a)].">
      <j1 rule="hyper" parents="37 15"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="265">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      z = y
    ]]></literal>
    <justification jstring="[para(172(a,1),37(a,1,2)),rewrite(181(3),243(5),181(5))].">
      <j1 rule="para" parents="172 37"/>
      <j2 rule="rewrite" parents="181 243 181"/>
    </justification>
  </clause>

  <clause id="269">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = (x @ z) @ y
    ]]></literal>
    <justification jstring="[hyper(38,a,17,a)].">
      <j1 rule="hyper" parents="38 17"/>
    </justification>
  </clause>

  <clause id="276">
    <literal><![CDATA[
      x * y != y * x
    ]]></literal>
    <literal><![CDATA[
      y @ x = c_0
    ]]></literal>
    <justification jstring="[para(181(a,1),38(a,1,2))].">
      <j1 rule="para" parents="181 38"/>
    </justification>
  </clause>

  <clause id="346">
    <literal><![CDATA[
      (x @ y) @ y = c_0
    ]]></literal>
    <justification jstring="[hyper(175,a,217,a),flip(a)].">
      <j1 rule="hyper" parents="175 217"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="347">
    <literal><![CDATA[
      (x @ y) * y = y * (x @ y)
    ]]></literal>
    <justification jstring="[para(217(a,1),4(a,1,2)),flip(a)].">
      <j1 rule="para" parents="217 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="357">
    <literal><![CDATA[
      (x @ y) * y * z = y * (x @ y) * z
    ]]></literal>
    <justification jstring="[para(346(a,1),14(a,1,2,2,1)),rewrite(172(3)),flip(a)].">
      <j1 rule="para" parents="346 14"/>
      <j2 rule="rewrite" parents="172"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="365">
    <literal><![CDATA[
      x * y * ((x @ y) @ y * x) = y * (x @ y) * x
    ]]></literal>
    <justification jstring="[back_rewrite(198),rewrite(357(8))].">
      <j1 rule="back_rewrite" parents="198"/>
      <j2 rule="rewrite" parents="357"/>
    </justification>
  </clause>

  <clause id="367">
    <literal><![CDATA[
      x @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(276,a,347,a)].">
      <j1 rule="hyper" parents="276 347"/>
    </justification>
  </clause>

  <clause id="404">
    <literal><![CDATA[
      (x @ y * x) * (y @ y * x) = c_0
    ]]></literal>
    <justification jstring="[hyper(175,a,68,a),flip(a)].">
      <j1 rule="hyper" parents="175 68"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="416">
    <literal><![CDATA[
      (x @ y) @ y * x = c_0
    ]]></literal>
    <justification jstring="[para(68(a,1),17(a,1)),rewrite(404(8))].">
      <j1 rule="para" parents="68 17"/>
      <j2 rule="rewrite" parents="404"/>
    </justification>
  </clause>

  <clause id="424">
    <literal><![CDATA[
      x * (y @ x) * y = y * x
    ]]></literal>
    <justification jstring="[back_rewrite(365),rewrite(416(3),181(2)),flip(a)].">
      <j1 rule="back_rewrite" parents="365"/>
      <j2 rule="rewrite" parents="416 181"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="448">
    <literal><![CDATA[
      x * (x @ y * x) = (x @ y) * x
    ]]></literal>
    <justification jstring="[hyper(73,a,14,a)].">
      <j1 rule="hyper" parents="73 14"/>
    </justification>
  </clause>

  <clause id="466">
    <literal><![CDATA[
      (x @ y) * x = x * (x @ y)
    ]]></literal>
    <justification jstring="[hyper(11,a,424,a),flip(a)].">
      <j1 rule="hyper" parents="11 424"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="495">
    <literal><![CDATA[
      x * (x @ y * x) = x * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(448),rewrite(466(5))].">
      <j1 rule="back_rewrite" parents="448"/>
      <j2 rule="rewrite" parents="466"/>
    </justification>
  </clause>

  <clause id="572">
    <literal><![CDATA[
      (x @ x * y) * y * (y @ x) = y
    ]]></literal>
    <justification jstring="[para(466(a,1),68(a,1,2)),rewrite(495(5))].">
      <j1 rule="para" parents="466 68"/>
      <j2 rule="rewrite" parents="495"/>
    </justification>
  </clause>

  <clause id="620">
    <literal><![CDATA[
      x @ x * y = x @ y
    ]]></literal>
    <justification jstring="[hyper(265,a,259,a),flip(a)].">
      <j1 rule="hyper" parents="265 259"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="638">
    <literal><![CDATA[
      x * (y @ x) * (x @ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(572),rewrite(620(2),357(4))].">
      <j1 rule="back_rewrite" parents="572"/>
      <j2 rule="rewrite" parents="620 357"/>
    </justification>
  </clause>

  <clause id="674">
    <literal><![CDATA[
      (x @ y) * (y @ x * y) = c_0
    ]]></literal>
    <justification jstring="[back_rewrite(183),rewrite(620(2))].">
      <j1 rule="back_rewrite" parents="183"/>
      <j2 rule="rewrite" parents="620"/>
    </justification>
  </clause>

  <clause id="734">
    <literal><![CDATA[
      (x @ y) * (y @ x) = c_0
    ]]></literal>
    <justification jstring="[hyper(175,a,638,a),flip(a)].">
      <j1 rule="hyper" parents="175 638"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="763">
    <literal><![CDATA[
      (x @ y) * (y @ x) * z = z
    ]]></literal>
    <justification jstring="[para(734(a,1),3(a,1,1)),rewrite(172(2)),flip(a)].">
      <j1 rule="para" parents="734 3"/>
      <j2 rule="rewrite" parents="172"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="808">
    <literal><![CDATA[
      (x @ y) @ (y @ x) = c_0
    ]]></literal>
    <justification jstring="[para(734(a,1),276(a,1)),rewrite(734(4)),xx(a)].">
      <j1 rule="para" parents="734 276"/>
      <j2 rule="rewrite" parents="734"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

  <clause id="1080">
    <literal><![CDATA[
      x @ y * x = x @ y
    ]]></literal>
    <justification jstring="[para(674(a,1),763(a,1,2)),rewrite(181(3)),flip(a)].">
      <j1 rule="para" parents="674 763"/>
      <j2 rule="rewrite" parents="181"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1116">
    <literal><![CDATA[
      (x @ y) @ z = z @ (y @ x)
    ]]></literal>
    <justification jstring="[para(763(a,1),1080(a,1,2)),rewrite(6(3),163(3),181(4),6(6),808(5),172(6))].">
      <j1 rule="para" parents="763 1080"/>
      <j2 rule="rewrite" parents="6 163 181 6 808 172"/>
    </justification>
  </clause>

  <clause id="1232">
    <literal><![CDATA[
      x * (D @ (F @ G)) * ((C @ (B @ A)) @ x) != (C @ (B @ A)) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(40),rewrite(1116(10),1116(18))].">
      <j1 rule="back_rewrite" parents="40"/>
      <j2 rule="rewrite" parents="1116 1116"/>
    </justification>
  </clause>

  <clause id="1500">
    <literal><![CDATA[
      (x @ y) @ (z @ y) = c_0
    ]]></literal>
    <justification jstring="[para(367(a,1),269(a,1,2)),rewrite(243(4)),flip(a)].">
      <j1 rule="para" parents="367 269"/>
      <j2 rule="rewrite" parents="243"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1514">
    <literal><![CDATA[
      x @ (y @ z) = c_0
    ]]></literal>
    <justification jstring="[para(269(a,2),734(a,1,1)),rewrite(1500(3),172(4))].">
      <j1 rule="para" parents="269 734"/>
      <j2 rule="rewrite" parents="1500 172"/>
    </justification>
  </clause>

  <clause id="1553">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(A)
    ]]></attribute>
    <justification jstring="[back_rewrite(1232),rewrite(1514(5),1514(6),166(3),181(3),181(2),1514(5),172(2)),xx(a)].">
      <j1 rule="back_rewrite" parents="1232"/>
      <j2 rule="rewrite" parents="1514 1514 166 181 181 1514 172"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
