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<b>let </b><font color="Maroon" title="c1">p</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> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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 href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">p</font>;<br/>
<a NAME="E1:59"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>, <a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <font color="Maroon" title="c1">p</font>, <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> , <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> ]
 
;<br/>
<div><i><font color="Green" title="E36">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">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> ;<br/><b>let </b><font color="Maroon" title="c3">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> ;<br/><b>let </b><font color="Maroon" title="c4">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> ;<br/><b>let </b><font color="Maroon" title="c5">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> ;<br/><b>assume </b><a NAME="E1:59_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> <font color="Maroon" title="c1">p</font>
 ;<br/><a NAME="E2:59_1"/>
<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#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T10" target="_self" title="CQC_SIM1:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E3:59_1"/>
( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <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>] implies <font color="Maroon">S<sub>1</sub></font>[<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>] )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T77" title="QC_LANG2:th.77">QC_LANG2:77</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E37">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">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> ;<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> ;<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> ;<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> ;<br/><b>assume </b><a NAME="E1:59_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="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c3">r</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" title="c1">p</font>
 ;<br/><a NAME="E2:59_2"/>
( <font color="Maroon" title="c2">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c3">r</font> &amp; <font color="Maroon" title="c3">r</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c3">r</font> )
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T11" target="_self" title="CQC_SIM1:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E3:59_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font>] implies ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c4">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="c2">q</font></span>)</span>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font>] ) )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T77" title="QC_LANG2:th.77">QC_LANG2:77</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="59_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">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> ;<br/><b>let </b><font color="Maroon" title="c3">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> ;<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> ;<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> ;<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> ;<br/><b>assume </b><a NAME="E1:59_3"/>
<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="c3">x</font>,<font color="Maroon" title="c2">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" title="c1">p</font>
 ;<br/><a NAME="E2:59_3"/>
<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#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T12" target="_self" title="CQC_SIM1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:59_3"/>
( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">K</font>,<font color="Maroon" title="c6">f</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c4">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1,<font color="Maroon" title="c5">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="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c6">f</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="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&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="c4">k</font></span>)</span></span>)</span>] )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T77" title="QC_LANG2:th.77">QC_LANG2:77</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:59"/>
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"><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>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <font color="Maroon" title="c1">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<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>]
 <b>from </b><i><a class="ref" href="cqc_sim1.html#S6" target="_self" title="CQC_SIM1:sch.6">CQC_SIM1:sch 6</a>(<a class="txt" href="cqc_sim1.html#E1:59"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="cqc_sim1.html#E2:59"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="cqc_sim1.html#E3:59"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="cqc_sim1.html#E4:59"><i><font color="Green" title="E38">A4</font></i></a>);</i><br/>


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