<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b2">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Olive" title="b2">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Olive" title="b2">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Olive" title="b2">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b2">Sub</font></span>)</span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Sub</font> being  <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Olive" title="b1">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Olive" title="b1">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">Sub</font></span>)</span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">Sub</font> be   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span></span>)</span></span><br/>





<b>set </b><font color="Maroon" title="c4">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span>}</span> ;<br/>
<a NAME="E1:105"/>
 <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Sub</font> <a href="substut1.html#K6" title="SUBSTUT1:func.6">|</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="substut1.html#D6" title="SUBSTUT1:def.6">SUBSTUT1:def 6</a></i>;<br/>
<a NAME="E2:105"/><b>then </b>
 <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E3:105"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E4:105"/><b>then </b><i><font color="Green" title="E84">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span>}</span>  &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c5">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:105_1"/><i><font color="Green" title="E85">A2</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:105_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">Sub</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:105"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="#E3:105_1"><i><font color="Green" title="E85">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E6:105_1"/><i><font color="Green" title="E86">A3</font></i>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c6">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">Sub</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">Sub</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 ;<br/><div><i><font color="Green" title="E87">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:105_1_1"/><b>then </b>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="#E4:105"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">y1</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E4:105_1_1"/><i><font color="Green" title="E88">A5</font></i>: 
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; not <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="ref" href="sublemma.html#D10" target="_self" title="SUBLEMMA:def.10">Def11</a></i>;<br/><b>thus </b><a NAME="E5:105_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:105_1"><i><font color="Green" title="E86">A3</font></i></a>, <a class="txt" href="#E4:105_1_1"><i><font color="Green" title="E88">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_1_2"/>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:105_1_2"/><b>then </b>
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font>
 <b>by </b><i><a class="txt" href="#E4:105"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">y1</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E4:105_1_2"/><i><font color="Green" title="E88">A6</font></i>: 
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y1</font> )
 <b>by </b><i><a class="ref" href="sublemma.html#D11" target="_self" title="SUBLEMMA:def.11">Def12</a></i>;<br/><b>thus </b><a NAME="E5:105_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:105_1"><i><font color="Green" title="E86">A3</font></i></a>, <a class="txt" href="#E4:105_1_2"><i><font color="Green" title="E88">A6</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E9:105_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:105_1"><i><font color="Green" title="E85">A2</font></i></a>, <a class="txt" href="#E7:105_1"><i><font color="Green" title="E87">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:105"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" title="SUBLEMMA:func.16">RSub1</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c3">Sub</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" title="SUBLEMMA:func.17">RSub2</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">Sub</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
