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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>assume </b><a NAME="E1:33"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">Sub</font> being  <a href="substut1.html#NM1">CQC_Substitution</a> holds   <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<font color="Olive">Sub</font></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">Sub</font> be   <a href="substut1.html#NM1">CQC_Substitution</a>;<br/>

<b>set </b><font color="Maroon">S1</font> = <span class="p1"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span>;<br/>
<b>set </b><font color="Maroon">S</font> = <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K6" target="_self">CFQ</a>  <a href="funct_2.html#K3">.</a> <span class="p3"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>;<br/>
<a NAME="E2:33"/><i><font color="Green">E44</font></i>: 
 <a href="substut2.html#K7" target="_self">QScope</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font> is <a href="substut1.html#V3">quantifiable</a>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E45</font></i>:  <a href="substut1.html#K24">Sub_All</a> <span class="p1">(<span class="default"><a href="substut2.html#K7" target="_self">QScope</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut2.html#K8" target="_self">Qsc</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span> = 
 <a href="sublemma.html#K10">CQCSub_All</a> <span class="p1">(<span class="default"><a href="substut2.html#K7" target="_self">QScope</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span>,<span class="p1">(<span class="default"><a href="substut2.html#K8" target="_self">Qsc</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span>
<b>by </b><i><a class="ref" href="sublemma.html#D6">SUBLEMMA:def 6</a></i>
<br/>.= 
<span class="p1"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span>
<b>by </b><i/>
;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span> is <a href="substut1.html#V7">Sub_universal</a>
 
<b>by </b><i>, <a class="ref" href="substut1.html#D28">SUBSTUT1:def 28</a></i>;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="substut1.html#K39">CQC_Sub</a> <span class="p1"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="sublemma.html#K12">CQCQuant</a> <span class="p1"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span>,<span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2">(<span class="default"><a href="sublemma.html#K11">CQCSub_the_scope_of</a> <span class="p3"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T29">SUBLEMMA:29</a></i>;<br/>
 <a href="sublemma.html#K11">CQCSub_the_scope_of</a> <span class="p1"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span> = 
 <a href="substut1.html#K31">Sub_the_scope_of</a> <span class="p1">(<span class="default"><a href="substut1.html#K24">Sub_All</a> <span class="p2">(<span class="default"><a href="substut2.html#K7" target="_self">QScope</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span>,<span class="p2">(<span class="default"><a href="substut2.html#K8" target="_self">Qsc</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="sublemma.html#D7">SUBLEMMA:def 7</a></i>
<br/>.= 
<span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K6" target="_self">CFQ</a>  <a href="funct_2.html#K3">.</a> <span class="p4"><a href="substut2.html#K2" target="_self">[</a><span class="default"><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><a href="substut2.html#K2" target="_self">]</a></span></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> <a href="sublemma.html#K9">`1</a> 
<b>by </b><i>, <a class="ref" href="substut1.html#T21">SUBSTUT1:21</a></i>
<br/>.= 
<span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="substut2.html#K6" target="_self">CFQ</a>  <a href="funct_2.html#K3">.</a> <span class="p3"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<a NAME="E7:33"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="substut1.html#K39">CQC_Sub</a> <span class="p1"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="sublemma.html#K12">CQCQuant</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10">CQCSub_All</a> <span class="p2">(<span class="default"><a href="substut2.html#K7" target="_self">QScope</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span>,<span class="p2">(<span class="default"><a href="substut2.html#K8" target="_self">Qsc</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K6" target="_self">CFQ</a>  <a href="funct_2.html#K3">.</a> <span class="p4"><a href="substut2.html#K2" target="_self">[</a><span class="default"><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><a href="substut2.html#K2" target="_self">]</a></span></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:33"/><i><font color="Green">E61</font></i>: 
( <span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K6" target="_self">CFQ</a>  <a href="funct_2.html#K3">.</a> <span class="p4"><a href="substut2.html#K2" target="_self">[</a><span class="default"><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><a href="substut2.html#K2" target="_self">]</a></span></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> is <a href="substut1.html#V3">quantifiable</a> &amp;  <a href="substut2.html#K7" target="_self">QScope</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="sublemma.html#K8">[</a><span class="default"><span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<span class="p3">(<span class="default"><a href="substut2.html#K6" target="_self">CFQ</a>  <a href="funct_2.html#K3">.</a> <span class="p4"><a href="substut2.html#K2" target="_self">[</a><span class="default"><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><a href="substut2.html#K2" target="_self">]</a></span></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>,<font color="Maroon">x</font></span><a href="sublemma.html#K8">]</a></span> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">y</font> =  <a href="substut1.html#K35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10">CQCSub_All</a> <span class="p3">(<span class="default"><a href="substut2.html#K7" target="_self">QScope</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span>,<span class="p3">(<span class="default"><a href="substut2.html#K8" target="_self">Qsc</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span>;<br/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span></span>)</span> = 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <span class="p2">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10">CQCSub_All</a> <span class="p5">(<span class="default"><a href="substut2.html#K7" target="_self">QScope</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span>,<span class="p5">(<span class="default"><a href="substut2.html#K8" target="_self">Qsc</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">Sub</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p3"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<span class="p4">(<span class="default"><a href="substut2.html#K6" target="_self">CFQ</a>  <a href="funct_2.html#K3">.</a> <span class="p5"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="sublemma.html#T32">SUBLEMMA:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p2">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p3"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">p</font>,<span class="p4">(<span class="default"><a href="substut2.html#K6" target="_self">CFQ</a>  <a href="funct_2.html#K3">.</a> <span class="p5"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="cqc_sim1.html#T18">CQC_SIM1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E10:33"/>
 <a href="cqc_sim1.html#K6">QuantNbr</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>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" 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#K2" target="_self">]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T18">CQC_SIM1:18</a></i>;<br/>


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