<?xml version="1.0"?>
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<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">S</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">xSQ</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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>;<br/>





<b>set </b><font color="Maroon">S1</font> =  <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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font>;<br/>
<b>set </b><font color="Maroon">z</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" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E1:53"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span> is <a href="substut1.html#V3">quantifiable</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <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>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span> )
 ;<br/>

<a NAME="E2:53"/><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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:53"/><b>then </b><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <span class="p1">(<span class="default"><span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span> <a href="substut1.html#K23">`2</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span> <a href="sublemma.html#K9" target="_self">`1</a> </span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">xSQ</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:53"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span> <a href="sublemma.html#K9" target="_self">`1</a> </span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E5:53"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="qc_lang1.html#K20">bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T8">QC_LANG2:8</a></i>;<br/>
<b>set </b><font color="Maroon">finSub</font> =  <a href="substut1.html#K7">RestrictSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span>;<br/>
<a NAME="E6:53"/><i><font color="Green">E59</font></i>: 
 <a href="substut1.html#K7">RestrictSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font>
 
<b>by </b><i>, , , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>set </b><font color="Maroon">n</font> =  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K11">Sub_Var</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E7:53"/>
 <a href="substut1.html#K12">NSub</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="substut1.html#K11">Sub_Var</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D11">SUBSTUT1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="substut1.html#K11">Sub_Var</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E9:53"/><i><font color="Green">E60</font></i>: 
 <a href="substut1.html#K13">upVar</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D12">SUBSTUT1:def 12</a></i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E61</font></i>: 
 <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="substut1.html#D11">SUBSTUT1:def 11</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green">E62</font></i>: 
 <a href="substut1.html#K11">Sub_Var</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K11">Sub_Var</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E12:53"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="substut1.html#K11">Sub_Var</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E13:53"/><b>then </b><i><font color="Green">E63</font></i>: 
not  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K11">Sub_Var</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E14:53"/>
 <a href="substut1.html#K11">Sub_Var</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="substut1.html#D10">SUBSTUT1:def 10</a></i>;<br/>
<a NAME="E15:53"/><b>then </b><i><font color="Green">E64</font></i>: 
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="substut1.html#K13">upVar</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:53"/><i><font color="Green">E65</font></i>: 
 <a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K11">Sub_Var</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E17:53"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="substut1.html#K11">Sub_Var</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<a NAME="E18:53"/><b>then </b><i><font color="Green">E66</font></i>: 
not  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><a href="substut1.html#K12">NSub</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E19:53"/><i><font color="Green">E67</font></i>: 
 <a href="substut1.html#K10">Dom_Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="qc_lang3.html#K2">x.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="substut1.html#D9">SUBSTUT1:def 9</a></i>;<br/>
<a NAME="E20:53"/><i><font color="Green">E68</font></i>: 
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <span class="p2">(<span class="default"><a href="substut1.html#K12">NSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p4"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E21:53"/>
<span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E22:53"/><b>then </b><i><font color="Green">E129</font></i>: 
not  <a href="qc_lang3.html#K2">x.</a> <span class="p1">(<span class="default"><a href="substut1.html#K13">upVar</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K9">Bound_Vars</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="qc_lang2.html#T8">QC_LANG2:8</a></i>;<br/>
<a NAME="E23:53"/>
 <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">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K32">@</a> <span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D35">SUBSTUT1:def 35</a></i>;<br/>
<b>hence </b><a NAME="E24:53"/>
( not  <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" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <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>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span> &amp; not  <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" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="substut1.html#K9">Bound_Vars</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="substut1.html#D36">SUBSTUT1:def 36</a></i>;<br/>


</div>
