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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:96_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font><br/> for <font color="Olive">ll</font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">k</font><br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Olive">P</font>,<font color="Olive">ll</font>,<font color="Olive">e</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">ll</font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1" target="_self">vSUB</a> ;<br/>







<b>assume </b><a NAME="E1:96_1_1"/>
 <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">e</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 ;<br/>

<a NAME="E2:96_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 
;<br/>
<a NAME="E3:96_1_1"/><i><font color="Green">E46</font></i>: 
 <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">e</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">S1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:96_1_1"/><i><font color="Green">E47</font></i>: 
(  <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> &amp; <font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a>  is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E6:96_1_1"/>
<font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">ll</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> 
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E7:96_1_1"/><b>then </b>
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K4">free_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T17">CQC_LANG:17</a></i>;<br/>
<a NAME="E8:96_1_1"/><b>then </b>
<font color="Maroon">ll</font> is   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T15">CQC_LANG:15</a></i>;<br/>
<b>hence </b><a NAME="E9:96_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K17" target="_self">Sub_P</a> <font color="Maroon">P</font>,<font color="Maroon">ll</font>,<font color="Maroon">e</font>]
 <b>by </b><i><a class="ref" href="substut1.html#D8" target="_self">Def8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:96_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>   st <font color="Olive">S</font> is <a href="substut1.html#V2" target="_self">Sub_VERUM</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>]
 
<b>by </b><i><a class="ref" href="substut1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:96_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Olive">S</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:96_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">S</font>]
 ;<br/>

<b>assume </b><a NAME="E2:96_1_2"/>
 <a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Maroon">S</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 ;<br/>

<a NAME="E3:96_1_2"/><b>then </b>
 <a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 
;<br/>
<b>then consider </b><font color="Maroon">S'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:96_1_2"/><i><font color="Green">E51</font></i>: 
(  <a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S'</font> &amp; <font color="Maroon">S'</font> <a href="substut1.html#K18" target="_self">`1</a>  is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  )
 ;<br/>
<a NAME="E6:96_1_2"/>
<font color="Maroon">S'</font> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E7:96_1_2"/><b>then </b>
<font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a>  is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> 
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T18">CQC_LANG:18</a></i>;<br/>
<a NAME="E8:96_1_2"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 
;<br/>
<b>hence </b><a NAME="E9:96_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K20" target="_self">Sub_not</a> <font color="Maroon">S</font>]
 <b>by </b><i><a class="ref" href="substut1.html#D8" target="_self">Def8</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:96_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">S1</font>, <font color="Olive">S2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>   st <font color="Olive">S1</font> <a href="substut1.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">S2</font> <a href="substut1.html#K19" target="_self">`2</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Olive">S1</font>,<font color="Olive">S2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> , <font color="Maroon">S2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:96_1_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">S1</font> <a href="substut1.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="substut1.html#K19" target="_self">`2</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">S1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">S2</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:96_1_3"/>
 <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 ;<br/>

<a NAME="E3:96_1_3"/><b>then </b>
 <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 
;<br/>
<b>then consider </b><font color="Maroon">S'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:96_1_3"/><i><font color="Green">E75</font></i>: 
(  <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S'</font> &amp; <font color="Maroon">S'</font> <a href="substut1.html#K18" target="_self">`1</a>  is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  )
 ;<br/>
<a NAME="E6:96_1_3"/>
 <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K19" target="_self">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:96_1_3"/><b>then </b>
<font color="Maroon">S'</font> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E8:96_1_3"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a>  is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a>  is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  )
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T19">CQC_LANG:19</a></i>;<br/>
<a NAME="E9:96_1_3"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">S1</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 
;<br/>
<a NAME="E10:96_1_3"/>
<font color="Maroon">S2</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:96_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font>]
 <b>by </b><i><a class="ref" href="substut1.html#D8" target="_self">Def8</a>, , <a class="txt" href="substut1.html#E10"><i><font color="Green">Lemma67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:96_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <br/> for <font color="Olive">SQ</font> being   <a href="substut1.html#M1" target="_self">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Olive">S</font>,<font color="Olive">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>  st <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Olive">S</font>,<font color="Olive">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> is <a href="substut1.html#V3" target="_self">quantifiable</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Olive">S</font>,<font color="Olive">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Olive">SQ</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">SQ</font> be    <a href="substut1.html#M1" target="_self">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>;<br/>





<b>assume </b><a NAME="E1:96_1_4"/><i><font color="Green">E81</font></i>: 
( <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> is <a href="substut1.html#V3" target="_self">quantifiable</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">S</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:96_1_4"/>
 <a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 ;<br/>

<a NAME="E3:96_1_4"/><b>then </b>
 <a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 
;<br/>
<b>then consider </b><font color="Maroon">S'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:96_1_4"/><i><font color="Green">E82</font></i>: 
(  <a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S'</font> &amp; <font color="Maroon">S'</font> <a href="substut1.html#K18" target="_self">`1</a>  is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  )
 ;<br/>
<a NAME="E6:96_1_4"/>
 <a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14">All</a> <span class="p3">(<span class="default"><span class="p4"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> <a href="substut1.html#K23" target="_self">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> <a href="substut1.html#K22" target="_self">`1</a> </span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">SQ</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:96_1_4"/><b>then </b>
<font color="Maroon">S'</font> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <span class="p1">(<span class="default"><span class="p2"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> <a href="substut1.html#K23" target="_self">`2</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> <a href="substut1.html#K22" target="_self">`1</a> </span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E8:96_1_4"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> <a href="substut1.html#K22" target="_self">`1</a> </span>)</span> <a href="substut1.html#K18" target="_self">`1</a>  is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> 
 
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T23">CQC_LANG:23</a></i>;<br/>
<a NAME="E9:96_1_4"/>
<span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span> <a href="substut1.html#K22" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E10:96_1_4"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a> 
 
<b>by </b><i><a class="txt" href="substut1.html#E10"><i><font color="Green">Lemma67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:96_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K24" target="_self">Sub_All</a> <span class="p1"><a href="substut1.html#K25" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="substut1.html#K25" target="_self">]</a></span>,<font color="Maroon">SQ</font>]
 <b>by </b><i><a class="ref" href="substut1.html#D8" target="_self">Def8</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:96_1"/>
for <font color="Olive">S</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>]
 
<b>from </b><i><a class="ref" href="substut1.html#S1" target="_self">SUBSTUT1:sch 1</a>(, , , , );<br/></i>
<b>hence </b><a NAME="E7:96_1"/>
for <font color="Olive">S</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38" target="_self">CQC-Sub-WFF</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">S</font>]
 ;<br/>


</div>
