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

<b>set </b><font color="Maroon">S</font> = <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> : <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> </span>}</span> ;<br/>
<a NAME="E1:54_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> : <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:54_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> : <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> </span>}</span> 
 ;<br/>

<a NAME="E2:54_1_1_1"/><b>then </b>
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:54_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> : <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> ;<br/>
<b>take </b><font color="Maroon">X</font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">A</font>;<br/>

<b>set </b><font color="Maroon">B</font> =  <a href="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>;<br/>
<a NAME="E3:54_1_1"/>
 <a href="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:54_1_1_2"/><i><font color="Green">E28</font></i>: 
 <a href="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:54_1_1_2"/>
<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="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T84">MCART_1:84</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_sim1.html#D12" target="_self">CQC_SIM1:def 12</a><br/>

<b>thus </b><a NAME="E3:54_1_1_2"/>
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> <br/> 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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Olive">q</font></span>)</span>,<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="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> holds <br/><span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">q</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="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, <a class="ref" href="mcart_1.html#T84">MCART_1:84</a></i>;<br/>

<b>thus </b><a NAME="E4:54_1_1_2"/>
for <font color="Olive">q</font>, <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <br/> 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"><span class="p2">(<span class="default"><font color="Olive">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">r</font></span>)</span>,<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="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> holds <br/>( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">q</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="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">r</font>,<span class="p2">(<span class="default"><font color="Olive">k</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="Olive">q</font></span>)</span></span>)</span>,<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="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> )
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, <a class="ref" href="mcart_1.html#T84">MCART_1:84</a></i>;<br/>

<b>let </b><font color="Maroon">q</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>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="E5:54_1_1_2"/>
<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">q</font></span>)</span>,<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="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>
 ;<br/>

<i><font color="Green">E29</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><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="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><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="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<a href="qc_lang1.html#K2">bound_QC-variables</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3"><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="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<a href="qc_lang1.html#K2">bound_QC-variables</a>  <a href="xboole_0.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="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="qc_lang1.html#K2">bound_QC-variables</a> 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/>
<a NAME="E7:54_1_1_2"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><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="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3"><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="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E8:54_1_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:54_1_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K2">rng</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">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E10:54_1_1_2"/>
<a href="qc_lang1.html#K2">bound_QC-variables</a>  <a href="xboole_0.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">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>;<br/>
<a NAME="E11:54_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3"><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="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E7:54_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="cqc_sim1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E12:54_1_1_2"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><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="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3"><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="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:54_1_1_2"/><b>then </b>
<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">k</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><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ff</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">k</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/>
<a NAME="E15:54_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<span class="p2">(<span class="default"><font color="Maroon">k</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>,<font color="Maroon">ff</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, <a class="ref" href="mcart_1.html#T84">MCART_1:84</a></i>;<br/>
<b>hence </b><a NAME="E16:54_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><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>,<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">k</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="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:54_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="subset_1.html#K2">[#]</a> <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<a NAME="E5:54_1_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><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> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_1_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:54_1_1_3"/><b>then </b>
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> )
 
;<br/>
<b>hence </b><a NAME="E3:54_1_1_3"/>
<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> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:54_1_1"/>
<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> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_sim1.html#D12" target="_self">CQC_SIM1:def 12</a><br/>

<b>thus </b><a NAME="E7:54_1_1"/>
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> <br/> 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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Olive">q</font></span>)</span>,<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> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">q</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> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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">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="E1:54_1_1_4"/><i><font color="Green">E41</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">q</font></span>)</span>,<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> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:54_1_1_4"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<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> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_1_4_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:54_1_1_4_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> )
 
;<br/>
<a NAME="E3:54_1_1_4_1"/>
<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">q</font></span>)</span>,<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> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1_1_4_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<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> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54_1_1_4"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<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> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E8:54_1_1"/>
for <font color="Olive">q</font>, <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <br/> 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"><span class="p2">(<span class="default"><font color="Olive">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">r</font></span>)</span>,<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> <font color="Maroon">X</font> holds <br/>( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">q</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> <font color="Maroon">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">r</font>,<span class="p2">(<span class="default"><font color="Olive">k</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="Olive">q</font></span>)</span></span>)</span>,<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> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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">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">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="E1:54_1_1_5"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font></span>)</span>,<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> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:54_1_1_5"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<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> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_1_5_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:54_1_1_5_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> )
 
;<br/>
<a NAME="E3:54_1_1_5_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font></span>)</span>,<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> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1_1_5_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<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> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54_1_1_5"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<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> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>

<a NAME="E4:54_1_1_5"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><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">k</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">q</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> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_1_5_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:54_1_1_5_2"/><b>then </b><i><font color="Green">E77</font></i>: 
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> )
 
;<br/>
<a NAME="E3:54_1_1_5_2"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font></span>)</span>,<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> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1_1_5_2"/>
<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">k</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">q</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> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_1_1_5"/>
<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">k</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">q</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> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E9:54_1_1"/>
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> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="qc_lang1.html#K2">bound_QC-variables</a> <br/> 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"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Olive">x</font>,<font color="Olive">q</font></span>)</span>,<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> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">q</font>,<span class="p2">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive">K</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">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="Olive">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="Olive">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.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="54_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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>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="E1:54_1_1_6"/><i><font color="Green">E78</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">q</font></span>)</span>,<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> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:54_1_1_6"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><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>,<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">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:54_1_1_6_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:54_1_1_6_1"/><b>then </b><i><font color="Green">E80</font></i>: 
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Olive">X</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font> )
 
;<br/>
<a NAME="E3:54_1_1_6_1"/>
<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">q</font></span>)</span>,<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> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:54_1_1_6_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><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>,<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">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:54_1_1_6"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">q</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">K</font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><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>,<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">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T2" target="_self">Th2</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">D</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<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#K4">:]</a></span>;<br/>

<b>assume </b><a NAME="E10:54_1_1"/>
<font color="Maroon">D</font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E11:54_1_1"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>hence </b><a NAME="E12:54_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>


</div>
