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<b>let </b><font color="Maroon">p</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">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">Sub</font> be   <a href="substut1.html#NM1">CQC_Substitution</a>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/>







<b>set </b><font color="Maroon">Sub1</font> = <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">Sub</font> = <font color="Maroon">Sub</font> as   <a href="substut1.html#NM1">CQC_Substitution</a> ;<br/>
<b>assume </b><a NAME="E2:11"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">S</font> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">S</font> <a href="sublemma.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 ;<br/>

<a NAME="E3:11"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K14">ExpandSub</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:11"/><i><font color="Green">E45</font></i>: 
(  <a href="qc_lang1.html#K20">bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="qc_lang1.html#K21">the_scope_of</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T8">QC_LANG2:8</a></i>;<br/>
<a NAME="E5:11"/>
 <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> is <a href="qc_lang1.html#V5">universal</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D20">QC_LANG1:def 20</a></i>;<br/>
<a NAME="E6:11"/><b>then </b>
 <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font> <a href="substut1.html#R1">PQSub</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="substut1.html#D14">SUBSTUT1:def 14</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:11"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p4">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p0">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>,<font color="Maroon">Sub</font> <a href="substut1.html#R1">PQSub</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span></span>)</span></span>)</span> )
 ;<br/>
<a NAME="E9:11"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K15">QSub</a> 
 
<b>by </b><i>, <a class="ref" href="substut1.html#D15">SUBSTUT1:def 15</a></i>;<br/>
<a NAME="E10:11"/><b>then </b><i><font color="Green">E59</font></i>: 
<a href="substut1.html#K15">QSub</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E11:11"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="sublemma.html#K8">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> <a href="sublemma.html#K9">`1</a> </span>)</span> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E12:11"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> <a href="substut1.html#K23">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E13:11"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="sublemma.html#K8">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> <a href="sublemma.html#K9">`1</a> </span>)</span> <a href="sublemma.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E14:11"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> is <a href="substut1.html#V3">quantifiable</a>
 
<b>by </b><i>, , , <a class="ref" href="substut1.html#D22">SUBSTUT1:def 22</a></i>;<br/>
<b>thus </b><a NAME="E15:11"/>
<span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> is <a href="substut1.html#V3">quantifiable</a>
 <b>by </b><i>, , , , <a class="ref" href="substut1.html#D22">SUBSTUT1:def 22</a></i>;<br/>

<b>reconsider </b><font color="Maroon">Sub</font> = <font color="Maroon">Sub</font> as    <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> <b>by </b><i>, , , , , <a class="ref" href="substut1.html#D23">SUBSTUT1:def 23</a></i>;<br/>
<b>take </b><font color="Maroon">S1</font> =  <a href="sublemma.html#K10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span>,<font color="Maroon">Sub</font>;<br/>

<a NAME="E17:11"/>
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K24">Sub_All</a> <span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span>,<font color="Maroon">Sub</font>
 
<b>by </b><i>, <a class="ref" href="sublemma.html#D6">SUBLEMMA:def 6</a></i>;<br/>
<b>hence </b><a NAME="E18:11"/>
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="substut2.html#K1" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<font color="Maroon">Sub</font></span><a href="substut2.html#K1" target="_self">]</a></span>
 <b>by </b><i>, , , <a class="ref" href="substut1.html#D24">SUBSTUT1:def 24</a></i>;<br/>


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