<?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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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/>
<b>thus </b><a NAME="E1:102"/>
 <a href="substut1.html#K2">@</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span> <a href="tarski.html#R1">c=</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#K3">\</a> <span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:102_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K2">@</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>
 ;<br/>

<a NAME="E2:102_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font>
 
<b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E3:102_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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="E4:102_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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="E5:102_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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 class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T96">RELAT_1:96</a></i>;<br/>
<a NAME="E6:102_1"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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 class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<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">c</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:102_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>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="E10:102_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">y1</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">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</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">y1</font> )
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>
 ;<br/><a NAME="E2:102_1_1"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:102_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="sublemma.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font>
 ;<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="E3:102_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">y</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">y</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:102_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="sublemma.html#T9" target="_self">Th9</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>
<a NAME="E13:102_1"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:102_1"/><b>then </b>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:102_1"/><b>then </b><i><font color="Green">E61</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T96">RELAT_1:96</a></i>;<br/>
<a NAME="E16:102_1"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:102_1"/><b>then </b>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E18:102_1"/><b>then </b>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T96">RELAT_1:96</a></i>;<br/>
<a NAME="E19:102_1"/><b>then </b><i><font color="Green">E62</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K1">\/</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E20:102_1"/>
 <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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E21:102_1"/><b>then </b>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_4.html#T32">FUNCT_4:32</a></i>;<br/>
<b>hence </b><a NAME="E22:102_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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#K3">\</a> <span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:102"/>
<span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="relset_1.html#K3">\</a> <span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="substut1.html#K2">@</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:102_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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#K3">\</a> <span class="p1">(<span class="default"><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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:102_2"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:102_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E5:102_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c</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> <font color="Maroon">Sub</font></span>)</span> &amp; <font color="Maroon">d</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">c</font> )
 
<b>by </b><i>, <a class="ref" href="sublemma.html#T9" target="_self">Th9</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">c</font> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> ;<br/>
<a NAME="E7:102_2"/>
 <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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:102_2"/><b>then </b>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K1">\/</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> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_4.html#T32">FUNCT_4:32</a></i>;<br/>
<a NAME="E9:102_2"/><b>then </b><i><font color="Green">E67</font></i>: 
( not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</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"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:102_2"/><b>then </b>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T96">RELAT_1:96</a></i>;<br/>
<a NAME="E11:102_2"/><b>then </b><i><font color="Green">E68</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></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/>
<a NAME="E12:102_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E13:102_2"/><b>then </b><i><font color="Green">E129</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T12" target="_self">Th12</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:102_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E15:102_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/>
<a NAME="E16:102_2"/><b>then </b>
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E17:102_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:102_2"/>
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T96">RELAT_1:96</a></i>;<br/>
<a NAME="E19:102_2"/><b>then </b><i><font color="Green">E148</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></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/>
<a NAME="E20:102_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E21:102_2"/><b>then </b><i><font color="Green">E149</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T16" target="_self">Th16</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:102_2"/><i><font color="Green">E150</font></i>: 
(  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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">z</font> )
 
<b>by </b><i><a class="ref" href="sublemma.html#T17" target="_self">Th17</a>, </i>;<br/>
<a NAME="E23:102_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">z</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">z</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T13" target="_self">Th13</a>, <a class="ref" href="sublemma.html#T18" target="_self">Th18</a>, , <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E24:102_2"/>
<font color="Maroon">z</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Sub</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</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="E26:102_2"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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> )
 <b>by </b><i><a class="ref" href="sublemma.html#T14" target="_self">Th14</a>, <a class="ref" href="sublemma.html#T15" target="_self">Th15</a>, <a class="ref" href="sublemma.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E27:102_2"/>
( <font color="Maroon">z</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, <a class="ref" href="sublemma.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E28:102_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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 class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E29:102_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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 class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></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/>
<a NAME="E30:102_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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="relat_1.html#T96">RELAT_1:96</a></i>;<br/>
<a NAME="E31:102_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> &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="E32:102_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font>
 
<b>by </b><i><a class="ref" href="substut1.html#D6">SUBSTUT1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E33:102_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K2">@</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>
 <b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/>


</div><b>end;</b></div>


</div>
