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<a NAME="E1:108_1"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="substut1.html#K19">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21">Sub_&amp;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:108_1_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="substut1.html#K19">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/>

<a NAME="E2:108_1_1"/><b>then </b>
 <a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K21">Sub_&amp;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="sublemma.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E3:108_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21">Sub_&amp;</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 <b>by </b><i><a class="txt" href="sublemma.html#E1:108"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sublemma.html#E1:108_1_1"><i><font color="Green">E92</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:108_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> st <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> is <a href="substut1.html#V3">quantifiable</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K24">Sub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>;<br/>





<b>assume </b><a NAME="E1:108_1_2"/><i><font color="Green">E92</font></i>: 
( <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> is <a href="substut1.html#V3">quantifiable</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] )
 ;<br/>

<a NAME="E2:108_1_2"/><b>then </b>
 <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K24">Sub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="sublemma.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:108_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K24">Sub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="sublemma.html#E1:108"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sublemma.html#E1:108_1_2"><i><font color="Green">E92</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:108_1"/><b>then </b><i><font color="Green">E92</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>6</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Olive">b<sub>5</sub></font><br/>for <font color="Olive">b<sub>7</sub></font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">b<sub>5</sub></font><br/>for <font color="Olive">b<sub>8</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1">vSUB</a>  holds <br/>( <font color="Maroon">P<sub>1</sub></font>[ <a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>8</sub></font>] &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="substut1.html#V2">Sub_VERUM</a> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">P<sub>1</sub></font>[ <a href="sublemma.html#K6" target="_self">Sub_not</a> <font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="substut1.html#K19">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] implies <font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21">Sub_&amp;</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] ) &amp; ( <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> is <a href="substut1.html#V3">quantifiable</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K24">Sub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Olive">b<sub>4</sub></font>] ) )
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:108"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sublemma.html#E1:108_1"><i><font color="Green">E91</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:108_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a>  holds <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="substut1.html#S5">SUBSTUT1:sch 5</a>(<a class="txt" href="sublemma.html#E3:108_1"><i><font color="Green">E92</font></i></a>);</i><br/>


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