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

<a NAME="E1:58_1"/><i><font color="Green" title="E38">A5</font></i>: 
 <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D13" target="_self" title="CQC_SIM1:def.13">Def13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c1">Y</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c2">X</font> = <span class="p1">(<span class="default"><a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> ;<br/>
<a NAME="E3:58_1"/>
<font color="Maroon" title="c2">X</font> <a href="cqc_sim1.html#R1" title="CQC_SIM1:pred.1">is_Sep-closed_on</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:58_1_1"/><i><font color="Green" title="E39">A6</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T31" target="_self" title="CQC_SIM1:th.31">Th31</a></i>;<br/>
<a NAME="E2:58_1_1"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:58_1_1"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="#E1:58_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_sim1.html#D12" target="_self" title="CQC_SIM1:def.12">CQC_SIM1:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b4">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">K</font>,<font color="Olive" title="b4">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">K</font>,<font color="Olive" title="b4">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) &amp; ( for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b5">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b2">r</font></span>)</span>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">q</font>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b2">r</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">q</font></span>)</span></span>)</span>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) ) &amp; ( for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b5">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">q</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) )</span><br/>

<b>thus </b><a NAME="E4:58_1_1"/>
for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b4">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">K</font>,<font color="Olive" title="b4">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">K</font>,<font color="Olive" title="b4">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b5">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b2">r</font></span>)</span>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">q</font>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b2">r</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">q</font></span>)</span></span>)</span>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) ) &amp; ( for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b5">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">q</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b3">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b2">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c5">K</font> be   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c6">f</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>







<b>assume </b><a NAME="E1:58_1_1_1"/><i><font color="Green" title="E40">A7</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E2:58_1_1_1"/><b>then </b><i><font color="Green" title="E41">A8</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:58_1_1_1"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E1:58_1_1_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c8">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c9">h</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Maroon" title="c10">L</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E5:58_1_1_1"/><i><font color="Green" title="E42">A9</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c10">L</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:58_1_1_1"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c10">L</font>,<font color="Maroon" title="c9">h</font>]
 ;<br/>
<a NAME="E7:58_1_1_1"/>
(  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">h</font> )
 
<b>by </b><i><a class="txt" href="#E5:58_1_1_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</a></i>;<br/>
<a NAME="E8:58_1_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font>]
 
<b>by </b><i><a class="txt" href="#E2:58"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E2:58_1_1_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="#E6:58_1_1_1"><i><font color="Green" title="E43">A10</font></i></a></i>;<br/>
<a NAME="E9:58_1_1_1"/><b>then </b>
( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span>  &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="#E1:58_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E2:58_1_1_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="cqc_sim1.html#D12" target="_self" title="CQC_SIM1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E10:58_1_1_1"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E5:58_1_1"/>
for <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b5">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b2">r</font></span>)</span>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">q</font>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b2">r</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">q</font></span>)</span></span>)</span>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b3">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b5">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">q</font>,<span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b3">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font></span>)</span>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/><b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b2">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/><b>let </b><font color="Maroon" title="c6">K</font> be   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/>( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/><b>let </b><font color="Maroon" title="c7">f</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies ( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> ) )</span><br/>









<b>assume </b><a NAME="E1:58_1_1_2"/><i><font color="Green" title="E40">A11</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<a NAME="E2:58_1_1_2"/><b>then </b><i><font color="Green" title="E41">A12</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:58_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E1:58_1_1_2"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c9">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c10">h</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Maroon" title="c11">L</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E5:58_1_1_2"/><i><font color="Green" title="E42">A13</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">L</font>,<font color="Maroon" title="c10">h</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <b>and </b><br/><a NAME="E6:58_1_1_2"/><i><font color="Green" title="E43">A14</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">L</font>,<font color="Maroon" title="c10">h</font>]
 ;<br/>
<a NAME="E7:58_1_1_2"/>
( <font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">h</font> )
 
<b>by </b><i><a class="txt" href="#E5:58_1_1_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</a></i>;<br/>
<a NAME="E8:58_1_1_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font>] )
 
<b>by </b><i><a class="txt" href="#E3:58"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="#E2:58_1_1_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="txt" href="#E6:58_1_1_2"><i><font color="Green" title="E43">A14</font></i></a></i>;<br/>
<a NAME="E9:58_1_1_2"/><b>then </b><i><font color="Green" title="E44">A15</font></i>: 
( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span>  &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span>  )
 
;<br/>
<a NAME="E10:58_1_1_2"/>
( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="#E1:58_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E2:58_1_1_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="cqc_sim1.html#D12" target="_self" title="CQC_SIM1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E11:58_1_1_2"/>
( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="#E9:58_1_1_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c3">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b4">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">K</font>,<font color="Olive" title="b4">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b3">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b2">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c6">K</font> be   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Olive" title="b1">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/><b>let </b><font color="Maroon" title="c7">f</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>









<b>assume </b><a NAME="E6:58_1_1"/><i><font color="Green" title="E40">A16</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E7:58_1_1"/><b>then </b><i><font color="Green" title="E41">A17</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:58_1_1"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E6:58_1_1"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c9">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c10">h</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Maroon" title="c11">L</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E10:58_1_1"/><i><font color="Green" title="E42">A18</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c6">K</font>,<font color="Maroon" title="c7">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">L</font>,<font color="Maroon" title="c10">h</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <b>and </b><br/><a NAME="E11:58_1_1"/><i><font color="Green" title="E43">A19</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">L</font>,<font color="Maroon" title="c10">h</font>]
 ;<br/>
<a NAME="E12:58_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E16"><i><font color="Green" title="E13">Lm1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c7">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E14:58_1_1"/>
(  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c6">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">L</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c10">h</font> )
 
<b>by </b><i><a class="txt" href="#E10:58_1_1"><i><font color="Green" title="E42">A18</font></i></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</a></i>;<br/>
<a NAME="E15:58_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c6">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c12">g</font>]
 
<b>by </b><i><a class="txt" href="#E4:58"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="#E7:58_1_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="txt" href="#E11:58_1_1"><i><font color="Green" title="E43">A19</font></i></a></i>;<br/>
<a NAME="E16:58_1_1"/><b>then </b><i><font color="Green" title="E44">A20</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span> 
 
;<br/>
<a NAME="E17:58_1_1"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="#E1:58_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E7:58_1_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="ref" href="cqc_sim1.html#D12" target="_self" title="CQC_SIM1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E18:58_1_1"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="#E16:58_1_1"><i><font color="Green" title="E44">A20</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:58_1"/><b>then </b><i><font color="Green" title="E39">A21</font></i>: 
 <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D13" target="_self" title="CQC_SIM1:def.13">Def13</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b3">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">K</font>,<font color="Olive" title="b3">f</font>]</span><br/><b>let </b><font color="Maroon" title="c4">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <br/> for <font color="Olive" title="b2">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Olive" title="b1">K</font>,<font color="Olive" title="b2">f</font>]</span><br/><b>let </b><font color="Maroon" title="c5">K</font> be   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Olive" title="b1">f</font>]</span><br/><b>let </b><font color="Maroon" title="c6">f</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font>] )</span><br/>







<b>assume </b><a NAME="E5:58_1"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font>]</span><br/>

<a NAME="E6:58_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Olive" title="b2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Olive" title="b3">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Olive" title="b4">K</font> is   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">k</font>,<font color="Olive" title="b4">K</font>,<font color="Olive" title="b3">f</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E4:58_1"><i><font color="Green" title="E39">A21</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c8">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c9">h</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Maroon" title="c10">L</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E8:58_1"/><i><font color="Green" title="E40">A22</font></i>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c10">L</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <b>and </b><br/><a NAME="E9:58_1"/><i><font color="Green" title="E41">A23</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c10">L</font>,<font color="Maroon" title="c9">h</font>]
 ;<br/>
<a NAME="E10:58_1"/>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c5">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">h</font> )
 
<b>by </b><i><a class="txt" href="#E8:58_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:58_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font>]
 <b>by </b><i><a class="txt" href="#E9:58_1"><i><font color="Green" title="E41">A23</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
