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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="qc_lang1.html#NM8">QC-formula</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <a href="qc_lang1.html#K2">bound_QC-variables</a>  =  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="qc_lang1.html#NM8">QC-formula</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">a<sub>1</sub></font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="xboole_0.html#K2">\/</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<a NAME="E1:64"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>, <a href="cqc_sim1.html#K12" target="_self">index</a> <font color="Maroon">c<sub>1</sub></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/>
<a NAME="E2:64"/><i><font color="Green">E39</font></i>: 
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>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <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>] holds <br/><font color="Maroon">S<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>3</sub></font>,<font color="Olive">b<sub>4</sub></font>]
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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><b>that </b><br/><a NAME="E1:64_1"/>
<span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</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">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:64_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <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>]
 ;<br/><a NAME="E3:64_1"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>;<br/><a NAME="E4:64_1"/><b>then </b>
(  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp;  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:64_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</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>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</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>2</sub></font></span>)</span>,<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#E2:64_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</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>3</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>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><b>that </b><br/><a NAME="E1:64_2"/>
<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>3</sub></font>,<font color="Maroon">c<sub>2</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">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:64_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</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>]
 ;<br/><a NAME="E3:64_2"/>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finsub_1.html#K2">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/><a NAME="E4:64_2"/><b>then </b>
 <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="cqc_sim1.html#E2:64_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T44">XBOOLE_1:44</a></i>;<br/><b>hence </b><a NAME="E5:64_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1,<font color="Maroon">c<sub>5</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>3</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>6</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>3</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>4</sub></font></span>)</span></span>)</span>]
 <b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:64"/>
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"><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>  <a href="cqc_sim1.html#K15" target="_self">SepQuadruples</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<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>3</sub></font>,<font color="Olive">b<sub>4</sub></font>]
 <b>from </b><i><a class="ref" href="cqc_sim1.html#S6" target="_self">CQC_SIM1:sch 6</a>(<a class="txt" href="cqc_sim1.html#E1:64"><i><font color="Green">E38</font></i></a>, <a class="txt" href="cqc_sim1.html#E2:64"><i><font color="Green">E39</font></i></a>, <a class="txt" href="cqc_sim1.html#E3:64"><i><font color="Green">E40</font></i></a>, <a class="txt" href="cqc_sim1.html#E4:64"><i><font color="Green">E41</font></i></a>);</i><br/>


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