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

<a NAME="E1:24"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <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="24_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_1"/>
 <a href="qc_lang1.html#K8">QC-WFF</a>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><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>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T21">QC_LANG1:21</a>, <a class="ref" href="qc_lang1.html#D9">QC_LANG1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E2:24_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <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>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> 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>
 ;<br/>

<b>thus </b><a NAME="E3:24"/>
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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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="24_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="E1:24_2"/>
<font color="Maroon">p</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">ll</font> <a href="hidden.html#R1">=</a> <span class="p1"><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>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T23">QC_LANG1:23</a></i>;<br/>
<b>hence </b><a NAME="E2:24_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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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>thus </b><a NAME="E4:24"/>
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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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="qc_lang1.html#D13">QC_LANG1:def 13</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>

<b>thus </b><a NAME="E5:24"/>
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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> 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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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="24_3"><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="E1:24_3"/>
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_3"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8">QC-WFF</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K1" target="_self">vSUB</a>  &amp; <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="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> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:24_3"/><i><font color="Green">E43</font></i>: 
 <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2"><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="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">p'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D14">QC_LANG1:def 14</a></i>;<br/>
<a NAME="E6:24_3"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2"><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="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">p'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2"><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>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D12">QC_LANG1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:24_3"/>
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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>thus </b><a NAME="E6:24"/>
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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> 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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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="24_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>, <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="E1:24_4"/><i><font color="Green">E45</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#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_4"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8">QC-WFF</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K1" target="_self">vSUB</a>  &amp; <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="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> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</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="E6:24_4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8">QC-WFF</a>  &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K1" target="_self">vSUB</a>  &amp; <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="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>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:24_4"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">p'</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<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">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> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">q'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D15">QC_LANG1:def 15</a></i>;<br/>
<a NAME="E9:24_4"/>
(  <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/>
<b>hence </b><a NAME="E10:24_4"/>
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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>thus </b><a NAME="E7:24"/>
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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span> 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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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="24_5"><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="E1:24_5"/>
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<a href="substut1.html#K1" target="_self">vSUB</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_5"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K8">QC-WFF</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K1" target="_self">vSUB</a>  &amp; <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="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> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:24_5"/><i><font color="Green">E50</font></i>: 
 <a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<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">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="p2"><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="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">p'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D16">QC_LANG1:def 16</a></i>;<br/>
<a NAME="E6:24_5"/>
<span class="p1">(<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">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="p2"><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="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">p'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<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">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="p2"><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>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D12">QC_LANG1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E7:24_5"/>
<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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="qc_lang1.html#K8">QC-WFF</a> ,<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>


</div>
