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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:60_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  implies <font color="Maroon" title="c1">x</font> <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"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1_1"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">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> , <font color="Maroon" title="c3">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> , <font color="Maroon" title="c4">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> , <font color="Maroon" title="c5">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> <b> such that </b><br/><a NAME="E3:60_1_1"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <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="c2">q</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T31" title="DOMAIN_1:th.31">DOMAIN_1:31</a></i>;<br/>
<div><i><font color="Green" title="E37">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_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="c2">q</font></span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">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> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 ;<br/><a NAME="E2:60_1_1_1"/><b>then </b>
 <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c2">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T36" target="_self" title="CQC_SIM1:th.36">Th36</a></i>;<br/><a NAME="E3:60_1_1_1"/><b>then </b>
 <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="ref" href="qc_lang2.html#T99" title="QC_LANG2:th.99">QC_LANG2:99</a></i>;<br/><b>hence </b><a NAME="E4:60_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="qc_lang1.html#T51" title="QC_LANG1:th.51">QC_LANG1:51</a>, <a class="ref" href="qc_lang1.html#D18" title="QC_LANG1:def.18">QC_LANG1:def 18</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E38">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c6">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> <b> such that </b><a NAME="E2:60_1_1_2"/><i><font color="Green" title="E39">A5</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="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c6">r</font></span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">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> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 ;<br/><a NAME="E3:60_1_1_2"/>
<font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c6">r</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E2:60_1_1_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="cqc_sim1.html#T36" target="_self" title="CQC_SIM1:th.36">Th36</a></i>;<br/><a NAME="E4:60_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="ref" href="qc_lang2.html#T99" title="QC_LANG2:th.99">QC_LANG2:99</a></i>;<br/><b>hence </b><a NAME="E5:60_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="qc_lang1.html#T51" title="QC_LANG1:th.51">QC_LANG1:51</a>, <a class="ref" href="qc_lang1.html#D19" title="QC_LANG1:def.19">QC_LANG1:def 19</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E39">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c6">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> , <font color="Maroon" title="c7">l</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> <b> such that </b><a NAME="E2:60_1_1_3"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">l</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="c6">r</font></span>)</span>
 <b>and </b><br/><a NAME="E3:60_1_1_3"/><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"><font color="Maroon" title="c6">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c7">l</font>,<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c5">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> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 ;<br/><a NAME="E4:60_1_1_3"/>
<font color="Maroon" title="c6">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E3:60_1_1_3"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="cqc_sim1.html#T36" target="_self" title="CQC_SIM1:th.36">Th36</a></i>;<br/><a NAME="E5:60_1_1_3"/><b>then </b>
<font color="Maroon" title="c6">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="ref" href="qc_lang2.html#T99" title="QC_LANG2:th.99">QC_LANG2:99</a></i>;<br/><b>hence </b><a NAME="E6:60_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="qc_lang1.html#T51" title="QC_LANG1:th.51">QC_LANG1:51</a>, <a class="ref" href="qc_lang1.html#D19" title="QC_LANG1:def.19">QC_LANG1:def 19</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c6">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> , <font color="Maroon" title="c7">l</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="c8">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> <b> such that </b><a NAME="E2:60_1_1_4"/>
( <font color="Maroon" title="c7">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c8">h</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&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="c7">l</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> )
 <b>and </b><br/><a NAME="E3:60_1_1_4"/><i><font color="Green" title="E40">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#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c7">l</font>,<font color="Maroon" title="c4">K</font>,<font color="Maroon" title="c8">h</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> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  or <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="c6">x</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<font color="Maroon" title="c7">l</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">K</font> <a href="setwiseo.html#K6" title="SETWISEO:func.6">\</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c8">h</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> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  )
 ;<br/><a NAME="E4:60_1_1_4"/>
 <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E3:60_1_1_4"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="cqc_sim1.html#T36" target="_self" title="CQC_SIM1:th.36">Th36</a></i>;<br/><a NAME="E5:60_1_1_4"/><b>then </b>
 <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="ref" href="qc_lang2.html#T99" title="QC_LANG2:th.99">QC_LANG2:99</a></i>;<br/><b>hence </b><a NAME="E6:60_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="qc_lang1.html#T51" title="QC_LANG1:th.51">QC_LANG1:51</a>, <a class="ref" href="qc_lang1.html#D20" title="QC_LANG1:def.20">QC_LANG1:def 20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:60_1_1"/>
<font color="Maroon" title="c1">x</font> <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"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E1:60_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="cqc_sim1.html#E3:60_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="cqc_sim1.html#E4:60_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="cqc_sim1.html#E5:60_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="cqc_sim1.html#E6:60_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="cqc_sim1.html#T23" target="_self" title="CQC_SIM1:th.23">Th23</a>, <a class="ref" href="cqc_sim1.html#T35" target="_self" title="CQC_SIM1:th.35">Th35</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:60_1"/>
( <font color="Maroon" title="c1">x</font> <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"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<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> implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  )
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T23" target="_self" title="CQC_SIM1:th.23">Th23</a>, <a class="ref" href="cqc_sim1.html#T31" target="_self" title="CQC_SIM1:th.31">Th31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:60"/>
 <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<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="p3">(<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></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


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