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

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">S</font>] </span>}</span> ;<br/>
<a NAME="E1:45_1"/>
not <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">S</font>] </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> ;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  <b>by </b><i>, <a class="ref" href="qc_lang1.html#D13">QC_LANG1:def 13</a></i>;<br/>
<a NAME="E3:45_1_1"/>
<font color="Maroon">V</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:45_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">V</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E5:45_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">S</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:45_1_1"/>
not <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">S</font>] </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">S</font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E3:45_1"/>
<font color="Maroon">X</font> is <a href="substut1.html#V1" target="_self">QC-Sub-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:45_1_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <a href="finseq_1.html#K13">*</a> </span>)</span>,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:45_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">S</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E3:45_1_2_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S</font>] )
 ;<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">e</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> <b> such that </b><br/><a NAME="E5:45_1_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</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">e</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:45_1_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K10">@</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D12">QC_LANG1:def 12</a></i>;<br/>
<a NAME="E7:45_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E8:45_1_2_1"/>
<font color="Maroon">x</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"><span class="p3"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <a href="finseq_1.html#K13">*</a> </span>)</span>,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1_2"/>
<font color="Maroon">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <a href="finseq_1.html#K13">*</a> </span>)</span>,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="substut1.html#D16" target="_self">SUBSTUT1:def 16</a><br/>

<b>thus </b><a NAME="E3:45_1_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font><br/> for <font color="Olive">ll</font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">k</font><br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a>  holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">P</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">ll</font></span>)</span>,<font color="Olive">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">ll</font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> ;<br/>







<a NAME="E1:45_1_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">e</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E2:45_1_2_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:45_1_2_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:45_1_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">P</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="qc_lang1.html#T23">QC_LANG1:23</a></i>;<br/>


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

<a NAME="E4:45_1_2"/>
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a>  holds  <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><a href="cqc_lang.html#K9">VERUM</a> ,<font color="Olive">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> ;<br/>

<b>set </b><font color="Maroon">V</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  <b>by </b><i>, <a class="ref" href="qc_lang1.html#D13">QC_LANG1:def 13</a></i>;<br/>
<a NAME="E2:45_1_2_3"/>
<font color="Maroon">V</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:45_1_2_3"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">V</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:45_1_2_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:45_1_2"/>
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a>  holds  <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Olive">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D13">QC_LANG1:def 13</a></i>;<br/>

<b>thus </b><a NAME="E6:45_1_2"/>
for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span><br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a>   st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">p</font></span>)</span>,<font color="Olive">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> ;<br/>



<b>assume </b><a NAME="E1:45_1_2_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">S</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E3:45_1_2_4"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">S</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">e</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S</font>] )
 ;<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">e'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> <b> such that </b><br/><a NAME="E5:45_1_2_4"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S</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">e'</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E6:45_1_2_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Maroon">S</font>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:45_1_2_4"/>
 <a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:45_1_2_4"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19" target="_self">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<a NAME="E9:45_1_2_4"/>
<font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E10:45_1_2_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p'</font></span>)</span>,<font color="Maroon">e'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E11:45_1_2_4"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">p'</font></span>)</span></span>)</span>,<font color="Maroon">e'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D14">QC_LANG1:def 14</a></i>;<br/>
<a NAME="E12:45_1_2_4"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E13:45_1_2_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#D12">QC_LANG1:def 12</a></i>;<br/>


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

<b>thus </b><a NAME="E7:45_1_2"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span><br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a>   st <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">p</font>,<font color="Olive">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">q</font>,<font color="Olive">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Olive">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Olive">q</font></span>)</span>,<font color="Olive">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>, <font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> ;<br/>



<b>assume </b><a NAME="E1:45_1_2_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">S1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E3:45_1_2_5"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">S1</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">e</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S1</font>] )
 ;<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">e1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> <b> such that </b><br/><a NAME="E5:45_1_2_5"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <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">e1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E6:45_1_2_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">S2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E8:45_1_2_5"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S2</font>] )
 ;<br/>
<b>consider </b><font color="Maroon">q'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <font color="Maroon">e2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> <b> such that </b><br/><a NAME="E10:45_1_2_5"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q'</font>,<font color="Maroon">e2</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E11:45_1_2_5"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E12:45_1_2_5"/><b>then </b><i><font color="Green">E132</font></i>: 
(  <a href="qc_lang1.html#K10">@</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="qc_lang1.html#K10">@</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D12">QC_LANG1:def 12</a></i>;<br/>
<a NAME="E13:45_1_2_5"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">S1</font> <a href="substut1.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E14:45_1_2_5"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">S1</font> <a href="substut1.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="substut1.html#K19" target="_self">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E15:45_1_2_5"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font>]
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E16:45_1_2_5"/>
 <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:45_1_2_5"/><b>then </b><i><font color="Green">E136</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K19" target="_self">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:45_1_2_5"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E19:45_1_2_5"/>
<font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E20:45_1_2_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5"><a href="domain_1.html#K1">[</a><span class="default">2,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , , , <a class="ref" href="qc_lang1.html#D15">QC_LANG1:def 15</a></i>;<br/>


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

<b>thus </b><a NAME="E8:45_1_2"/>
for <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span><br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">p</font>,<span class="p2">(<span class="default"><a href="substut1.html#K15" target="_self">QSub</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p0"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font></span>)</span>,<font color="Olive">e</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font></span>)</span>,<font color="Olive">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_6"><b>proof </b></a><div class="add">

<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">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> ;<br/>





<b>assume </b><a NAME="E1:45_1_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut1.html#K15" target="_self">QSub</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p0"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">S</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E3:45_1_2_6"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut1.html#K15" target="_self">QSub</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p0"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S</font>] )
 ;<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">e'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> <b> such that </b><br/><a NAME="E5:45_1_2_6"/><i><font color="Green">E139</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut1.html#K15" target="_self">QSub</a>  <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p1"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p0"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <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">e'</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:45_1_2_6"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <a href="substut1.html#K15" target="_self">QSub</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E7:45_1_2_6"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">S</font> <a href="substut1.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <a href="substut1.html#K15" target="_self">QSub</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E8:45_1_2_6"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E9:45_1_2_6"/>
 <a href="qc_lang1.html#K10">@</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> 
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D12">QC_LANG1:def 12</a></i>;<br/>
<a NAME="E10:45_1_2_6"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">S</font> <a href="substut1.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <a href="substut1.html#K15" target="_self">QSub</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="qc_lang1.html#D16">QC_LANG1:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:45_1_2_6"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>
 ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i/>;<br/>
<a NAME="E14:45_1_2_6"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">B</font> <a href="substut1.html#K22" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">B</font> <a href="substut1.html#K23" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E15:45_1_2_6"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">B</font> is <a href="substut1.html#V3" target="_self">quantifiable</a>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as    <a href="substut1.html#M1" target="_self">second_Q_comp</a> of <font color="Maroon">B</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E17:45_1_2_6"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K24" target="_self">Sub_All</a> <font color="Maroon">B</font>,<font color="Maroon">e</font>]
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E18:45_1_2_6"/>
 <a href="substut1.html#K24" target="_self">Sub_All</a> <font color="Maroon">B</font>,<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:45_1_2_6"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14">All</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="substut1.html#K23" target="_self">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="substut1.html#K22" target="_self">`1</a> </span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E20:45_1_2_6"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="substut1.html#K22" target="_self">`1</a> </span>)</span> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E21:45_1_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">p'</font></span>)</span></span>)</span>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="substut1.html#T1" target="_self">Th1</a>, <a class="ref" href="qc_lang1.html#D16">QC_LANG1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E22:45_1_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p5"><a href="domain_1.html#K1">[</a><span class="default">3,0</span><a href="domain_1.html#K1">]</a></span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#D12">QC_LANG1:def 12</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E4:45_1"/><b>then </b><i><font color="Green">E150</font></i>: 
 <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">F</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/>

<a NAME="E5:45_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">S</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E7:45_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">S</font>] )
 ;<br/>
<b>thus </b><a NAME="E8:45_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F</font>]
 <b>by </b><i/>;<br/>


</div>
