<?xml version="1.0"?>
<div class="add">

<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">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<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>assume </b><a NAME="E1:70"/><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>
 ;<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>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E2:70"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S</font> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a>  <a href="substut1.html#K14">ExpandSub</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>,<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>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">finSub</font> =  <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>;<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E58</font></i>: 
<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="E2:70_1"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13">upVar</a> <span class="p5">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon">xSQ</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/><a NAME="E3:70_1"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13">upVar</a> <span class="p5">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon">xSQ</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p2">(<span class="default"><a href="substut1.html#K13">upVar</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<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>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T45">CQC_LANG:45</a></i>;<br/><b>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13">upVar</a> <span class="p5">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><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></span>)</span>,<font color="Maroon">xSQ</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>thus </b><a NAME="E5:70_1"/>
 <a href="relat_1.html#K1">dom</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> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_1_1"/>
not  <a href="relat_1.html#K1">dom</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> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:70_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>set </b><font color="Maroon">q</font> =  <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>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> ;<br/><a NAME="E4:70_1_1_1"/>
 <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> <a href="hidden.html#R1">=</a> <font color="Maroon">xSQ</font> <a href="substut1.html#K6">|</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D6">SUBSTUT1:def 6</a></i>;<br/><a NAME="E5:70_1_1_1"/><b>then </b>
 <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E6:70_1_1_1"/><b>then </b>
 <a href="substut1.html#K2">@</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:70_1_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</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>,<span class="p4">(<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></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E8:70_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K4">dom</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>,<span class="p4">(<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></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E9:70_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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>,<span class="p4">(<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></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:70_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E10:70_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="sublemma.html#E3:70_1_1_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E12:70_1_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Maroon">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/><b>thus </b><a NAME="E13:70_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sublemma.html#E3:70_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sublemma.html#E8:70_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:70_1_1"/>
 <a href="relat_1.html#K1">dom</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> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:70_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</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>,<span class="p4">(<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></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:70_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:70_1"/><b>then </b><i><font color="Green">E64</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>,<span class="p4">(<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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">F</font> <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>,<span class="p4">(<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></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T32">FUNCT_4:32</a></i>;<br/><a NAME="E8:70_1"/>
<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><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" 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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D1">VALUAT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:70_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><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" 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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E11:70_1"/><i><font color="Green">E66</font></i>: 
 <a href="relset_1.html#K5">rng</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>,<span class="p4">(<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></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E12:70_1"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="substut1.html#K13">upVar</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>,<span class="p0">(<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>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/><a NAME="E13:70_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K2">x.</a> <span class="p3">(<span class="default"><a href="substut1.html#K13">upVar</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>,<span class="p0">(<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>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/><a NAME="E14:70_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<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></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<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></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T10">FUNCT_7:10</a></i>;<br/><a NAME="E15:70_1"/>
<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#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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:70"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sublemma.html#E3:70"><i><font color="Green">E57</font></i></a>, <a class="ref" href="substut1.html#D13">SUBSTUT1:def 13</a></i>;<br/><a NAME="E16:70_1"/><b>then </b>
 <a href="substut1.html#K2">@</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span> <a href="hidden.html#R1">=</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E17:70_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<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></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:70_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="sublemma.html#E7:70_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E18:70_1"/>
<font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:70_1_2"/><i><font color="Green">E146</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E2:70_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E3:70_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/>
<a NAME="E4:70_1_2"/>
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_1_2_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<a NAME="E2:70_1_2_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E11:70_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="sublemma.html#E14:70_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:70_1_2_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<a NAME="E4:70_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E12:70_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="sublemma.html#E14:70_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:70_1_2_1"/><b>then </b><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E6:70_1_2_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <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> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<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>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E2:70_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="sublemma.html#T2" target="_self">Th2</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<a NAME="E7:70_1_2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i>, <a class="txt" href="sublemma.html#E3:70"><i><font color="Green">E57</font></i></a>, <a class="ref" href="sublemma.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E8:70_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:70_1_2"/>
<font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E11:70_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="sublemma.html#E12:70_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E19:70_1"/>
 <a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E10:70_1"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2"/><i><font color="Green">E152</font></i>: 
not <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="E2:70_2"/><i><font color="Green">E153</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/><a NAME="E3:70_2"/><i><font color="Green">E154</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T45">CQC_LANG:45</a></i>;<br/><b>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>thus </b><a NAME="E5:70_2"/>
 <a href="relat_1.html#K1">dom</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> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_2_1_1"/>
not  <a href="relat_1.html#K1">dom</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> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:70_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</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>,<span class="p4">(<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></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:70_2_1_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</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>,<span class="p4">(<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></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:70_2_1_1"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</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; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>set </b><font color="Maroon">q</font> =  <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>;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> ;<br/><a NAME="E6:70_2_1_1"/>
 <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> <a href="hidden.html#R1">=</a> <font color="Maroon">xSQ</font> <a href="substut1.html#K6">|</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D6">SUBSTUT1:def 6</a></i>;<br/><a NAME="E7:70_2_1_1"/><b>then </b>
 <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E8:70_2_1_1"/><b>then </b>
 <a href="substut1.html#K2">@</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E9:70_2_1_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</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>,<span class="p4">(<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></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><a NAME="E10:70_2_1_1"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="relset_1.html#K4">dom</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>,<span class="p4">(<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></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:70_2_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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>,<span class="p4">(<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></span>)</span>
 <b>by </b><i>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E12:70_2_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y1</font> where <font color="Olive">y1</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Olive">y1</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y1</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E14:70_2_1_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <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> &amp; <font color="Maroon">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/><b>thus </b><a NAME="E15:70_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:70_2_1"/>
 <a href="relat_1.html#K1">dom</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> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:70_2"/><b>then </b>
 <a href="relset_1.html#K4">dom</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>,<span class="p4">(<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></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:70_2"/><b>then </b><i><font color="Green">E159</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>,<span class="p4">(<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></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">F</font> <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>,<span class="p4">(<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></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T32">FUNCT_4:32</a></i>;<br/><a NAME="E8:70_2"/>
<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><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" 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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <a href="qc_lang1.html#K2">bound_QC-variables</a> ,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D1">VALUAT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E10:70_2"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><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" 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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E11:70_2"/><i><font color="Green">E161</font></i>: 
 <a href="relset_1.html#K5">rng</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>,<span class="p4">(<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></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E12:70_2"/><i><font color="Green">E162</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/><a NAME="E13:70_2"/><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<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></span>)</span> <a href="funct_4.html#K1">+*</a> <font color="Maroon">F</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<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></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_7.html#T10">FUNCT_7:10</a></i>;<br/><a NAME="E14:70_2"/>
<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#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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:70"><i><font color="Green">E55</font></i></a>, <a class="txt" href="sublemma.html#E3:70"><i><font color="Green">E57</font></i></a>, <a class="ref" href="substut1.html#D13">SUBSTUT1:def 13</a></i>;<br/><a NAME="E15:70_2"/><b>then </b>
 <a href="substut1.html#K2">@</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span> <a href="hidden.html#R1">=</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> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E16:70_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<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></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E17:70_2"/>
<font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:70_2_2"/><i><font color="Green">E165</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E2:70_2_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E3:70_2_2"/><b>then </b><i><font color="Green">E166</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/>
<a NAME="E4:70_2_2"/>
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_2_2_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<a NAME="E2:70_2_2_1"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:70_2_2_1"/><b>then </b><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<a NAME="E4:70_2_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:70_2_2_1"/><b>then </b><i><font color="Green">E170</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E6:70_2_2_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="sublemma.html#T2" target="_self">Th2</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<a NAME="E7:70_2_2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i>, <a class="txt" href="sublemma.html#E3:70"><i><font color="Green">E57</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:70_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:70_2_2"/>
<font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:70_2"/>
 <a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:70"/>
(  <a href="sublemma.html#K3" target="_self">Val_S</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32">@</a> <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></span>)</span></span>)</span> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</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> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="sublemma.html#E2:70"><i><font color="Green">E56</font></i></a></i>;<br/>


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