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<a NAME="E1:114_1"/><span class="lab"><font color="Green" title="E89">A2</font></span>: 
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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <br/> for <font color="Olive" title="b3">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">S</font>] holds <br/><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="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b2">S</font>,<font color="Olive" title="b1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b3">xSQ</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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> <br/> for <font color="Olive" title="b2">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] holds <br/><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="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b2">xSQ</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">xSQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>  st <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">S</font>] holds <br/><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="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b1">xSQ</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c3">xSQ</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">S</font>] implies <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="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font>] )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:114_1_1"/><span class="lab"><font color="Green" title="E90">A3</font></span>: 
<span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a> 
 <span class="kw">and </span><br/><a NAME="E2:114_1_1"/><span class="lab"><font color="Green" title="E91">A4</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">S</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font>]</span><br/>

<a NAME="E3:114_1_1"/>
 <a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K24" title="SUBSTUT1:func.24">Sub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_1"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="ref" href="sublemma.html#D6" target="_self" title="SUBLEMMA:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:114_1_1"/>
<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="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c3">xSQ</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E1:114_1_1"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="txt" href="#E2:114_1_1"><span class="lab"><font color="Green" title="E91">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:114_1"/>
for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">S9</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>   st <font color="Olive" title="b1">S</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">S9</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">S9</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">S9</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">S9</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">S</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">S9</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">S</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">S9</font>] implies <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">S</font>,<font color="Maroon" title="c2">S9</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:114_1_2"/><span class="lab"><font color="Green" title="E90">A5</font></span>: 
<font color="Maroon" title="c1">S</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">S9</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> 
 <span class="kw">and </span><br/><a NAME="E2:114_1_2"/><span class="lab"><font color="Green" title="E91">A6</font></span>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">S</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">S9</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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">S</font>,<font color="Maroon" title="c2">S9</font>]</span><br/>

<a NAME="E3:114_1_2"/>
 <a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">S9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">S9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114_1_2"><span class="lab"><font color="Green" title="E90">A5</font></span></a>, <a class="ref" href="sublemma.html#D4" target="_self" title="SUBLEMMA:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:114_1_2"/>
<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">S</font>,<font color="Maroon" title="c2">S9</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E1:114_1_2"><span class="lab"><font color="Green" title="E90">A5</font></span></a>, <a class="txt" href="#E2:114_1_2"><span class="lab"><font color="Green" title="E91">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:114_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A7</font></span>: 
for <font color="Olive" title="b1">S</font>, <font color="Olive" title="b2">S9</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> <br/> for <font color="Olive" title="b3">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b4">SQ</font> being   <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span><br/> for <font color="Olive" title="b5">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="b6">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b5">k</font><br/> for <font color="Olive" title="b7">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b5">k</font><br/> for <font color="Olive" title="b8">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 <br/> ( <font color="Maroon">P<sub>1</sub></font>[ <a href="sublemma.html#K4" title="SUBLEMMA:func.4">Sub_P</a> <font color="Olive" title="b7">P</font>,<font color="Olive" title="b6">ll</font>,<font color="Olive" title="b8">e</font>] &amp; ( <font color="Olive" title="b1">S</font> is  <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] ) &amp; ( <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] implies <font color="Maroon">P<sub>1</sub></font>[ <a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Olive" title="b1">S</font>] ) &amp; ( <font color="Olive" title="b1">S</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">S9</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">S9</font>] implies <font color="Maroon">P<sub>1</sub></font>[ <a href="substut1.html#K21" title="SUBSTUT1:func.21">Sub_&amp;</a> <font color="Olive" title="b1">S</font>,<font color="Olive" title="b2">S9</font>] ) &amp; ( <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is  <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] implies <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="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Olive" title="b1">S</font>,<font color="Olive" title="b3">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Olive" title="b4">SQ</font>] ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E1:114_1"><span class="lab"><font color="Green" title="E89">A2</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:114_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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="substut1.html#S5" title="SUBSTUT1:sch.5">SUBSTUT1:sch 5</a>(<a class="txt" href="#E3:114_1"><span class="lab"><font color="Green" title="E90">A7</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
