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

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<a NAME="E1:68"/><i><font color="Green">E28</font></i>: 
<a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <a href="cqc_lang.html#K9">VERUM</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="funcop_1.html#K2">--&gt;</a> <a href="cqc_lang.html#K9">VERUM</a> 
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ] <b>means </b>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">K</font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> <br/> for <font color="Olive">f</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Olive">k</font>,<font color="Olive">K</font>,<font color="Olive">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font> holds <br/><font color="Olive">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">k</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>;<br/>
<a NAME="E2:68"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9">VERUM</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>





<b>assume </b><a NAME="E1:68_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:68_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T7">QC_LANG3:7</a>, <a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/>
<b>hence </b><a NAME="E3:68_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">k</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="qc_lang3.html#T7">QC_LANG3:7</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">l</font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>;<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>thus </b><a NAME="E1:68_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">l</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>





<b>assume </b><a NAME="E1:68_2_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">l</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 ;<br/>

<b>set </b><font color="Maroon">fl</font> = <font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font>;<br/>
<a NAME="E2:68_2_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K5" target="_self">ATOMIC</a> <font color="Maroon">P</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">m</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:68_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> = 
<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>
;<br/>
<b>thus </b><a NAME="E2:68_2_1_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1_1_2"><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="E1:68_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:68_2_1_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:68_2_1_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E6:68_2_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:68_2_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E8:68_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:68_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i>, , , </i>;<br/>


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

<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="E3:68_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:68_2_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 ;<br/>
<a NAME="E6:68_2_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:68_2_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:68_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:68_2_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E10:68_2_1_1"/>
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E11:68_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">l</font></span>)</span></span>)</span> = 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T9">QC_LANG3:9</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K1">variables_in</a> <font color="Maroon">l</font>,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T6">QC_LANG3:6</a></i>
<br/>.= 
 <a href="qc_lang3.html#K1">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span>,<a href="qc_lang1.html#K2">bound_QC-variables</a> 
<b>by </b><i/>
<br/>.= 
 <a href="qc_lang1.html#K22">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T6">QC_LANG3:6</a></i>
<br/>.= 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="cqc_sim1.html#K4" target="_self">*</a> <font color="Maroon">l</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T9">QC_LANG3:9</a></i>
;<br/>
<b>hence </b><a NAME="E5:68_2_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E77</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>reconsider </b><font color="Maroon">g</font> = <a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <font color="Maroon">r</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/><a NAME="E2:68_3"/><i><font color="Green">E78</font></i>: 
<a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="cqc_sim1.html#K1" target="_self">NEGATIVE</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E3:68_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">r</font>]
 ;<br/><b>thus </b><a NAME="E4:68_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>





<b>assume </b><a NAME="E1:68_3_1"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 ;<br/>

<b>set </b><font color="Maroon">mf</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E3:68_3_1"/><i><font color="Green">E81</font></i>: 
(  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font></span>)</span> &amp;  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r'</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/>
<a NAME="E4:68_3_1"/>
( <span class="p1">(<span class="default"><a href="cqc_sim1.html#K1" target="_self">NEGATIVE</a> <font color="Maroon">g</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r'</font> &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">m</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E5:68_3_1"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:68_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">r</font>]
 <b>and </b><br/><a NAME="E2:68_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>]
 ;<br/><b>thus </b><a NAME="E3:68_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">g</font> = <a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <font color="Maroon">r</font>, <font color="Maroon">h</font> = <a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <font color="Maroon">s</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>





<b>assume </b><a NAME="E2:68_4_1"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 ;<br/>

<b>set </b><font color="Maroon">mf</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">m</font>,<font color="Maroon">f</font>, <font color="Maroon">s'</font> = <font color="Maroon">h</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="cqc_sim1.html#K6" target="_self">QuantNbr</a> <font color="Maroon">r</font></span>)</span></span>)</span>,<font color="Maroon">f</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E4:68_4_1"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K2" target="_self">CON</a> <font color="Maroon">g</font>,<font color="Maroon">h</font>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K6" target="_self">QuantNbr</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">m</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:68_4_1"/>
( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">m</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font> &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" target="_self">QuantNbr</a> <font color="Maroon">r</font></span>)</span></span>)</span>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="txt" href="cqc_sim1.html#E3:68"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E6:68_4_1"/><b>then </b><i><font color="Green">E87</font></i>: 
( <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r'</font> &amp; <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s'</font> )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span>)</span> = 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r'</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T153">RELAT_1:153</a></i>
<br/>.= 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">r'</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s'</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>
<br/>.= 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>assume </b><a NAME="E1:68_5"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">r</font>]
 ;<br/><b>thus </b><a NAME="E2:68_5"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_5_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">g</font> = <a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <font color="Maroon">r</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>





<b>assume </b><a NAME="E2:68_5_1"/><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>,<font color="Maroon">m</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:68_5_1"/>
<font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">m</font></span>)</span></span>)</span> is   <a href="funct_2.html#NM1">Function</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/>;<br/>
<b>then reconsider </b><font color="Maroon">fm</font> = <font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">m</font></span>)</span></span>)</span> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <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="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r''</font> = <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">fm</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E6:68_5_1"/><i><font color="Green">E91</font></i>: 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r</font></span>)</span> <a href="finsub_1.html#K2">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/>
<a NAME="E7:68_5_1"/><b>then </b><i><font color="Green">E92</font></i>: 
not  <a href="qc_lang3.html#K2">x.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r</font></span>)</span> <a href="finsub_1.html#K2">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:68_5_1"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K3" target="_self">UNIVERSAL</a> <font color="Maroon">x</font>,<font color="Maroon">g</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">m</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">r''</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:68_5_1"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="setwiseo.html#K5">\/</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="cqc_sim1.html#T3" target="_self">Th3</a></i>;<br/>
<b>thus </b><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span></span>)</span> = 
<font color="Maroon">fm</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r</font></span>)</span> <a href="finsub_1.html#K2">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fm</font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">m</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">r''</font></span>)</span> <a href="finsub_1.html#K2">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">m</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">r''</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>
<br/>.= 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K8" target="_self">SepFunc</a>  <a href="cqc_sim1.html#K7" target="_self">.</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E7:68"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>]
 
<b>from </b><i><a class="ref" href="cqc_sim1.html#S5" target="_self">CQC_SIM1:sch 5</a>(, , , , );<br/></i>
<a NAME="E8:68"/><i><font color="Green">E98</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K13" target="_self">id</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>thus </b> <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K13" target="_self">id</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T162">FUNCT_1:162</a></i>
<br/>.= 
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K14" target="_self">SepVar</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


</div>
