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

<a NAME="E1:29_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K38">CQC-Sub-WFF</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38">CQC-Sub-WFF</a> 
 ;<br/><b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <font color="Maroon">Sub</font> being   <a href="substut1.html#NM1">CQC_Substitution</a><b> such that </b><br/><a NAME="E3:29_1_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">Sub</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, <a class="ref" href="substut1.html#T8">SUBSTUT1:8</a></i>;<br/><div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p</font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><b>set </b><font color="Maroon">b</font> =  <a href="substut1.html#K14">ExpandSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span>;<br/><a NAME="E2:29_1_1_1_1_1"/>
<font color="Maroon">p</font>,<font color="Maroon">Sub</font> <a href="substut1.html#R1">PQSub</a>  <a href="substut1.html#K14">ExpandSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="substut1.html#D14">SUBSTUT1:def 14</a></i>;<br/><a NAME="E3:29_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">Sub</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="substut1.html#K14">ExpandSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K15">QSub</a> 
 <b>by </b><i><a class="ref" href="substut1.html#D15">SUBSTUT1:def 15</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="substut1.html#K15">QSub</a> 
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_1_1_2"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">p</font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><b>set </b><font color="Maroon">b</font> =  <a href="xboole_0.html#K1">{}</a> ;<br/><a NAME="E2:29_1_1_1_1_2"/>
<font color="Maroon">p</font>,<font color="Maroon">Sub</font> <a href="substut1.html#R1">PQSub</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="substut1.html#D14">SUBSTUT1:def 14</a></i>;<br/><a NAME="E3:29_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">Sub</font></span><a href="domain_1.html#K1">]</a></span>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K15">QSub</a> 
 <b>by </b><i><a class="ref" href="substut1.html#D15">SUBSTUT1:def 15</a></i>;<br/><b>hence </b><a NAME="E4:29_1_1_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="substut1.html#K15">QSub</a> 
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:29_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="substut1.html#K15">QSub</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:29_1_1_1"/><b>then </b>
 <a href="substut1.html#K38">CQC-Sub-WFF</a>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <a href="substut1.html#K15">QSub</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K38">CQC-Sub-WFF</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:29_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="substut1.html#K1">vSUB</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_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="E1:29_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a> </span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:29_1_1_2"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a> 
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E5:29_1_1_2"/>
<a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a>  <a href="tarski.html#R1">c=</a>  <a href="substut1.html#K15">QSub</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <font color="Maroon">Sub</font> being   <a href="substut1.html#NM1">CQC_Substitution</a>, <font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:29_1_1_2"/><i><font color="Green">E75</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">Sub</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">p</font>,<font color="Maroon">Sub</font> <a href="substut1.html#R1">PQSub</a> <font color="Maroon">b1</font> )
 <b>by </b><i>, <a class="ref" href="substut1.html#D15">SUBSTUT1:def 15</a></i>;<br/>
<div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_2_1"/>
<font color="Maroon">p</font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><a NAME="E2:29_1_1_2_1"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K14">ExpandSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="substut1.html#D14">SUBSTUT1:def 14</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_2_1"/>
<font color="Maroon">b</font> is   <a href="substut1.html#NM1">CQC_Substitution</a>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_1_1_2_2"/>
not <font color="Maroon">p</font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><a NAME="E2:29_1_1_2_2"/><b>then </b>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="substut1.html#D14">SUBSTUT1:def 14</a></i>;<br/><a NAME="E3:29_1_1_2_2"/><b>then </b>
<font color="Maroon">b1</font> is   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="qc_lang1.html#K2">bound_QC-variables</a> , <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#T56">PARTFUN1:56</a></i>;<br/><a NAME="E4:29_1_1_2_2"/><b>then </b>
<font color="Maroon">b1</font> is   <a href="substut1.html#NM1">CQC_Substitution</a>
 <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a>, <a class="ref" href="substut1.html#D1">SUBSTUT1:def 1</a></i>;<br/><b>hence </b><a NAME="E5:29_1_1_2_2"/>
<font color="Maroon">b</font> is   <a href="substut1.html#NM1">CQC_Substitution</a>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:29_1_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K1">vSUB</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_1"/>
<a href="substut1.html#K15">QSub</a>  <a href="relat_1.html#K7">|</a> <a href="substut1.html#K38">CQC-Sub-WFF</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> , <a href="substut1.html#K1">vSUB</a> 
 <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>


</div>
