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

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">Sub</font> be   <a href="substut1.html#NM1">CQC_Substitution</a>;<br/>





<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Sub</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Sub</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> ;<br/>
<a NAME="E1:103"/>
 <a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Sub</font> <a href="substut1.html#K6">|</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Sub</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Sub</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="substut1.html#D6">SUBSTUT1:def 6</a></i>;<br/>
<a NAME="E2:103"/><b>then </b>
 <a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Sub</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Sub</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E3:103"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Sub</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Sub</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E4:103"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Sub</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Sub</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span>  &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <font color="Maroon">p</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:103_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:103_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Sub</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Sub</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="sublemma.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E6:103_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Sub</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Sub</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/><div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:103_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K16" target="_self">RSub1</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E4:103_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; not <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:103_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, <a class="ref" href="sublemma.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:103_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K17" target="_self">RSub2</a> <font color="Maroon">p</font>,<font color="Maroon">Sub</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E4:103_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:103_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, <a class="ref" href="sublemma.html#T12" target="_self">Th12</a>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:103_1"/>
contradiction
 <b>by </b><i><a class="ref" href="sublemma.html#T9" target="_self">Th9</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:103"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div>
