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

<a NAME="E1:60_1"/><i><font color="Green">E40</font></i>: 
 <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() <a href="cqc_sim1.html#R1" target="_self">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">Def13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="numbers.html#K5">NAT</a> ,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="zfmisc_1.html#K4">:]</a></span> ;<br/>
<a NAME="E3:60_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_sim1.html#R1" target="_self">is_Sep-closed_on</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:60_1_1"/><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K13" target="_self">id</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E2:60_1_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K13" target="_self">id</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:60"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K13" target="_self">id</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> </span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E1:60_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_sim1.html#D12" target="_self">CQC_SIM1:def 12</a><br/>

<b>thus </b><a NAME="E4:60_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> <br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a>  st <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>







<b>assume </b><a NAME="E1:60_1_1_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:60_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:60_1_1_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:60_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>9</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , <font color="Maroon">c<sub>10</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E5:60_1_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="mcart_1.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:60_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>]
 ;<br/>
<a NAME="E7:60_1_1_1"/>
(  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E5:60_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mcart_1.html#T33">MCART_1:33</a></i>;<br/>
<a NAME="E8:60_1_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>]
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E2:60"><i><font color="Green">E37</font></i></a>, <a class="txt" href="cqc_sim1.html#E2:60_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="cqc_sim1.html#E6:60_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E9:60_1_1_1"/><b>then </b>
( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span>  &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:60_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cqc_sim1.html#E2:60_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cqc_sim1.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E10:60_1_1_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>thus </b><a NAME="E5:60_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a>  st <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" target="_self">QuantNbr</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>









<b>assume </b><a NAME="E1:60_1_1_2"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:60_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:60_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:60_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>10</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , <font color="Maroon">c<sub>11</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E5:60_1_1_2"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="mcart_1.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:60_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 ;<br/>
<a NAME="E7:60_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E5:60_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mcart_1.html#T33">MCART_1:33</a></i>;<br/>
<a NAME="E8:60_1_1_2"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" target="_self">QuantNbr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>] )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E3:60"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cqc_sim1.html#E2:60_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="cqc_sim1.html#E6:60_1_1_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E9:60_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span>  &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" target="_self">QuantNbr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span>  )
 
;<br/>
<a NAME="E10:60_1_1_2"/>
( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" target="_self">QuantNbr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:60_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cqc_sim1.html#E2:60_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cqc_sim1.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E11:60_1_1_2"/>
( <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="cqc_sim1.html#K6" target="_self">QuantNbr</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E9:60_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>









<b>assume </b><a NAME="E6:60_1_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E7:60_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:60_1_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E6:60_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>10</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , <font color="Maroon">c<sub>11</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E10:60_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="mcart_1.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E11:60_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 ;<br/>
<a NAME="E12:60_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> is   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K2">bound_QC-variables</a> , <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E16"><i><font color="Green">Lemma13</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E14:60_1_1"/>
(  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E10:60_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="mcart_1.html#T33">MCART_1:33</a></i>;<br/>
<a NAME="E15:60_1_1"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>12</sub></font>]
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E4:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cqc_sim1.html#E7:60_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="cqc_sim1.html#E11:60_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E16:60_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="setwiseo.html#K5">\/</a> <span class="p3"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> 
 
;<br/>
<a NAME="E17:60_1_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E1:60_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cqc_sim1.html#E7:60_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cqc_sim1.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E18:60_1_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E16:60_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:60_1"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D13" target="_self">Def13</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/>







<b>assume </b><a NAME="E5:60_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E6:60_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="mcart_1.html#K4">]</a></span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , B is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , B is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , B is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>3</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E4:60_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> , <font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>9</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> , <font color="Maroon">c<sub>10</sub></font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> <b> such that </b><br/><a NAME="E8:60_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="mcart_1.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E9:60_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font>]
 ;<br/>
<a NAME="E10:60_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="cqc_sim1.html#E8:60_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="mcart_1.html#T33">MCART_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:60_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>]
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E9:60_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div>
