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<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">p</font>;<br/>
<a NAME="E1:61"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>, <a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">p</font>, <a href="setwiseo.html#K1">{}.</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> , <a href="cqc_sim1.html#K13" target="_self">id</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ]
 
;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>assume </b><a NAME="E1:61_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:61_1"/>
<font color="Maroon">q</font> <a href="qc_lang2.html#R2">is_subformula_of</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:61_1"/>
( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font>,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font>] )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T77">QC_LANG2:77</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>assume </b><a NAME="E1:61_2"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:61_2"/>
( <font color="Maroon">q</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:61_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font>,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font>] implies ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">k</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">q</font></span>)</span>,<font color="Maroon">K</font>,<font color="Maroon">f</font>] ) )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T77">QC_LANG2:77</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">K</font> be   <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a> ;<br/><b>assume </b><a NAME="E1:61_3"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span>,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:61_3"/>
<font color="Maroon">q</font> <a href="qc_lang2.html#R2">is_subformula_of</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:61_3"/>
( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">q</font>,<font color="Maroon">k</font>,<font color="Maroon">K</font>,<font color="Maroon">f</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">q</font>,<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">K</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <font color="Maroon">k</font></span>)</span></span>)</span>] )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T77">QC_LANG2:77</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:61"/>
for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <br/> for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">K</font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of <a href="qc_lang1.html#K2">bound_QC-variables</a> <br/> for <font color="Olive">f</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<a href="qc_lang1.html#K2">bound_QC-variables</a>   st <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Olive">q</font>,<font color="Olive">k</font>,<font color="Olive">K</font>,<font color="Olive">f</font></span><a href="mcart_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>,<font color="Olive">k</font>,<font color="Olive">K</font>,<font color="Olive">f</font>]
 <b>from </b><i><a class="ref" href="cqc_sim1.html#S6" target="_self">CQC_SIM1:sch 6</a>(<a class="ref" href="cqc_sim1.html#T4" target="_self">Th4</a>, <a class="txt" href="cqc_sim1.html#E4:61"><i><font color="Green">E38</font></i></a>, , );</i><br/>


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