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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ] <b>means </b>( $1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a>  implies <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:96_1"/><i><font color="Green" title="E60">A2</font></i>: 
for <font color="Olive" title="b1">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="b2">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font><br/> for <font color="Olive" title="b3">ll</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Olive" title="b1">k</font><br/> for <font color="Olive" title="b4">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Olive" title="b2">P</font>,<font color="Olive" title="b3">ll</font>,<font color="Olive" title="b4">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" title="c1">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="c2">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>;<br/><b>let </b><font color="Maroon" title="c3">ll</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c1">k</font>;<br/><b>let </b><font color="Maroon" title="c4">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> ;<br/>







<b>assume </b><a NAME="E1:96_1_1"/>
 <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 ;<br/>

<a NAME="E2:96_1_1"/><b>then </b><i><font color="Green" title="E61">A3</font></i>: 
 <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 
;<br/>
<a NAME="E3:96_1_1"/><i><font color="Green" title="E62">A4</font></i>: 
 <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">ll</font></span>)</span>,<font color="Maroon" title="c4">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="substut1.html#T9" target="_self" title="SUBSTUT1:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">S1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:96_1_1"/><i><font color="Green" title="E63">A5</font></i>: 
(  <a href="substut1.html#K17" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S1</font> &amp; <font color="Maroon" title="c5">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is    <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>by </b><i><a class="txt" href="substut1.html#E2:96_1_1"><i><font color="Green" title="E61">A3</font></i></a></i>;<br/>
<a NAME="E6:96_1_1"/>
<font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">ll</font> is    <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>by </b><i><a class="txt" href="substut1.html#E3:96_1_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="txt" href="substut1.html#E5:96_1_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">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" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">ll</font> &amp; <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K4" title="QC_LANG1:func.4">free_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">ll</font> &amp; <font color="Maroon" title="c3">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T17" title="CQC_LANG:th.17">CQC_LANG:17</a></i>;<br/>
<a NAME="E8:96_1_1"/><b>then </b>
<font color="Maroon" title="c3">ll</font> is   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T15" title="CQC_LANG:th.15">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" title="SUBSTUT1:func.17">Sub_P</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">ll</font>,<font color="Maroon" title="c4">e</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:96"><i><font color="Green" title="E59">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:96_1"/><i><font color="Green" title="E61">A6</font></i>: 
for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>   st <font color="Olive" title="b1">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>]
 
<b>by </b><i><a class="txt" href="substut1.html#E1:96"><i><font color="Green" title="E59">A1</font></i></a></i>;<br/>
<a NAME="E3:96_1"/><i><font color="Green" title="E62">A7</font></i>: 
for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Olive" title="b1">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" title="c1">S</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:96_1_2"/><i><font color="Green" title="E63">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">S</font>]
 ;<br/>

<b>assume </b><a NAME="E2:96_1_2"/>
 <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 ;<br/>

<a NAME="E3:96_1_2"/><b>then </b>
 <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c2">S'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:96_1_2"/><i><font color="Green" title="E64">A9</font></i>: 
(  <a href="substut1.html#K20" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S'</font> &amp; <font color="Maroon" title="c2">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is    <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/>
<a NAME="E6:96_1_2"/>
<font color="Maroon" title="c2">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:96_1_2"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E7:96_1_2"/><b>then </b>
<font color="Maroon" title="c1">S</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is    <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>by </b><i><a class="txt" href="substut1.html#E5:96_1_2"><i><font color="Green" title="E64">A9</font></i></a>, <a class="ref" href="cqc_lang.html#T18" title="CQC_LANG:th.18">CQC_LANG:18</a></i>;<br/>
<a NAME="E8:96_1_2"/><b>then </b>
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">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" title="SUBSTUT1:func.20">Sub_not</a> <font color="Maroon" title="c1">S</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:96"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="substut1.html#E1:96_1_2"><i><font color="Green" title="E63">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:96_1"/><i><font color="Green" title="E63">A10</font></i>: 
for <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>   st <font color="Olive" title="b1">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">S2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">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" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:96_1_3"/><i><font color="Green" title="E64">A11</font></i>: 
( <font color="Maroon" title="c1">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">S1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">S2</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:96_1_3"/>
 <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 ;<br/>

<a NAME="E3:96_1_3"/><b>then </b>
 <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c3">S'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:96_1_3"/><i><font color="Green" title="E65">A12</font></i>: 
(  <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S'</font> &amp; <font color="Maroon" title="c3">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is    <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/>
<a NAME="E6:96_1_3"/>
 <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:96_1_3"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="substut1.html#D21" target="_self" title="SUBSTUT1:def.21">Def21</a></i>;<br/>
<a NAME="E7:96_1_3"/><b>then </b>
<font color="Maroon" title="c3">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:96_1_3"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E8:96_1_3"/><b>then </b><i><font color="Green" title="E66">A13</font></i>: 
( <font color="Maroon" title="c1">S1</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is    <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>  &amp; <font color="Maroon" title="c2">S2</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is    <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>by </b><i><a class="txt" href="substut1.html#E5:96_1_3"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="cqc_lang.html#T19" title="CQC_LANG:th.19">CQC_LANG:19</a></i>;<br/>
<a NAME="E9:96_1_3"/><b>then </b><i><font color="Green" title="E67">A14</font></i>: 
<font color="Maroon" title="c1">S1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 
;<br/>
<a NAME="E10:96_1_3"/>
<font color="Maroon" title="c2">S2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 
<b>by </b><i><a class="txt" href="substut1.html#E8:96_1_3"><i><font color="Green" title="E66">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:96_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:96"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="substut1.html#E1:96_1_3"><i><font color="Green" title="E64">A11</font></i></a>, <a class="txt" href="substut1.html#E9:96_1_3"><i><font color="Green" title="E67">A14</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:96_1"/><i><font color="Green" title="E64">A15</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b2">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> <br/> for <font color="Olive" title="b3">SQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>  st <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">S</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Olive" title="b3">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" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c3">SQ</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>;<br/>





<b>assume </b><a NAME="E1:96_1_4"/><i><font color="Green" title="E65">A16</font></i>: 
( <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">S</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:96_1_4"/>
 <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 ;<br/>

<a NAME="E3:96_1_4"/><b>then </b>
 <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c4">S'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E5:96_1_4"/><i><font color="Green" title="E66">A17</font></i>: 
(  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S'</font> &amp; <font color="Maroon" title="c4">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is    <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/>
<a NAME="E6:96_1_4"/>
 <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <span class="p3">(<span class="default"><span class="p4"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a> </span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">SQ</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:96_1_4"><i><font color="Green" title="E65">A16</font></i></a>, <a class="ref" href="substut1.html#D24" target="_self" title="SUBSTUT1:def.24">Def24</a></i>;<br/>
<a NAME="E7:96_1_4"/><b>then </b>
<font color="Maroon" title="c4">S'</font> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <span class="p1">(<span class="default"><span class="p2"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> <a href="substut1.html#K23" title="SUBSTUT1:func.23">`2</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a> </span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:96_1_4"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E8:96_1_4"/><b>then </b><i><font color="Green" title="E67">A18</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a> </span>)</span> <a href="substut1.html#K18" title="SUBSTUT1:func.18">`1</a>  is    <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>by </b><i><a class="txt" href="substut1.html#E5:96_1_4"><i><font color="Green" title="E66">A17</font></i></a>, <a class="ref" href="cqc_lang.html#T23" title="CQC_LANG:th.23">CQC_LANG:23</a></i>;<br/>
<a NAME="E9:96_1_4"/>
<span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span> <a href="substut1.html#K22" title="SUBSTUT1:func.22">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E10:96_1_4"/><b>then </b>
<font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> 
 
<b>by </b><i><a class="txt" href="substut1.html#E8:96_1_4"><i><font color="Green" title="E67">A18</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" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="substut1.html#K25" title="SUBSTUT1:func.25">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="substut1.html#K25" title="SUBSTUT1:func.25">]</a></span>,<font color="Maroon" title="c3">SQ</font>]
 <b>by </b><i><a class="txt" href="substut1.html#E1:96"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="substut1.html#E1:96_1_4"><i><font color="Green" title="E65">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:96_1"/>
for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>]
 
<b>from </b><i><a class="ref" href="substut1.html#S1" target="_self" title="SUBSTUT1:sch.1">SUBSTUT1:sch 1</a>(<a class="txt" href="substut1.html#E1:96_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="substut1.html#E2:96_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="substut1.html#E3:96_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="txt" href="substut1.html#E4:96_1"><i><font color="Green" title="E63">A10</font></i></a>, <a class="txt" href="substut1.html#E5:96_1"><i><font color="Green" title="E64">A15</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:96_1"/>
for <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>]
 ;<br/>


</div>
