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<div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c4">S</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c5">xSQ</font> be    <a href="substut1.html#M1" title="SUBSTUT1:mode.1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>;<br/>









<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E54">A1</font></i>: 
<span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span> is <a href="substut1.html#V3" title="SUBSTUT1:attr.3">quantifiable</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">S1</font> =  <a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font>;<br/>
<b>set </b><font color="Maroon" title="c7">z</font> =  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c8">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>;<br/>

<a NAME="E2:70"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K14" title="SUBSTUT1:func.14">ExpandSub</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:70"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="sublemma.html#T42" target="_self" title="SUBLEMMA:th.42">Th42</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">finSub</font> =  <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font>;<br/>
<div><i><font color="Green" title="E56">A3</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" title="E57">A4</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>
 ;<br/><a NAME="E2:70_1"/><i><font color="Green" title="E58">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p5">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p0">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/><a NAME="E3:70_1"/><i><font color="Green" title="E59">A6</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p5">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p0">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p2">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T45" title="CQC_LANG:th.45">CQC_LANG:45</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">F</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p4">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p5">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p0">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><b>thus </b><a NAME="E5:70_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:70_1_1_1"/><i><font color="Green" title="E60">A7</font></i>: 
( <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>set </b><font color="Maroon" title="c12">q</font> =  <a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c13">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> ;<br/><a NAME="E4:70_1_1_1"/>
 <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">xSQ</font> <a href="substut1.html#K6" title="SUBSTUT1:func.6">|</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D6" title="SUBSTUT1:def.6">SUBSTUT1:def 6</a></i>;<br/><a NAME="E5:70_1_1_1"/><b>then </b>
 <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E6:70_1_1_1"/><b>then </b>
 <a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:70_1_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E8:70_1_1_1"/><b>then </b><i><font color="Green" title="E61">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E9:70_1_1_1"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:70_1_1_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E10:70_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="sublemma.html#E8:70_1_1_1"><i><font color="Green" title="E61">A8</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E12:70_1_1_1"/><i><font color="Green" title="E62">A9</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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_1_1"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="sublemma.html#E12:70_1_1_1"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:70_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:70_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:70_1"/><b>then </b><i><font color="Green" title="E60">A10</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T32" title="FUNCT_4:th.32">FUNCT_4:32</a></i>;<br/><a NAME="E8:70_1"/>
<font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D1" title="VALUAT_1:def.1">VALUAT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:70_1"/><i><font color="Green" title="E61">A11</font></i>: 
( <font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E11:70_1"/><i><font color="Green" title="E62">A12</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E10:70_1"><i><font color="Green" title="E61">A11</font></i></a></i>;<br/><a NAME="E12:70_1"/><i><font color="Green" title="E63">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p3">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p4">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/><a NAME="E13:70_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p3">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p4">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/><a NAME="E14:70_1"/><b>then </b><i><font color="Green" title="E64">A14</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E10:70_1"><i><font color="Green" title="E61">A11</font></i></a>, <a class="txt" href="sublemma.html#E11:70_1"><i><font color="Green" title="E62">A12</font></i></a>, <a class="ref" href="funct_7.html#T10" title="FUNCT_7:th.10">FUNCT_7:10</a></i>;<br/><a NAME="E15:70_1"/>
<font color="Maroon" title="c4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:70"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="sublemma.html#E1:70_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="substut1.html#D13" title="SUBSTUT1:def.13">SUBSTUT1:def 13</a></i>;<br/><a NAME="E16:70_1"/><b>then </b>
 <a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E17:70_1"/><b>then </b><i><font color="Green" title="E65">A15</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E7:70_1"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="sublemma.html#E14:70_1"><i><font color="Green" title="E64">A14</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E18:70_1"/>
<font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">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" title="E66">A16</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E2:70_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">F</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E10:70_1"><i><font color="Green" title="E61">A11</font></i></a>, <a class="txt" href="sublemma.html#E12:70_1"><i><font color="Green" title="E63">A13</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E3:70_1_2"/><b>then </b><i><font color="Green" title="E67">A17</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/>
<a NAME="E4:70_1_2"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c12">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_1_2_1"/><i><font color="Green" title="E68">A18</font></i>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span>
 ;<br/>

<a NAME="E2:70_1_2_1"/><i><font color="Green" title="E69">A19</font></i>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:70_1_2"><i><font color="Green" title="E66">A16</font></i></a>, <a class="txt" href="sublemma.html#E1:70_1_2_1"><i><font color="Green" title="E68">A18</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:70_1_2_1"/><b>then </b><i><font color="Green" title="E70">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>;<br/>
<a NAME="E4:70_1_2_1"/>
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E3:70_1_2"><i><font color="Green" title="E67">A17</font></i></a>, <a class="txt" href="sublemma.html#E1:70_1_2_1"><i><font color="Green" title="E68">A18</font></i></a>, <a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:70_1_2_1"/><b>then </b><i><font color="Green" title="E71">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E6:70_1_2_1"/>
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <span class="p1">(<span class="default"><a href="substut1.html#K13" title="SUBSTUT1:func.13">upVar</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E3:70_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="sublemma.html#E2:70_1_2_1"><i><font color="Green" title="E69">A19</font></i></a>, <a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>;<br/>
<a NAME="E7:70_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:70"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="sublemma.html#E1:70_1"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="sublemma.html#T52" target="_self" title="SUBLEMMA:th.52">Th52</a></i>;<br/>
<b>hence </b><a NAME="E8:70_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:70_1_2_1"><i><font color="Green" title="E70">A20</font></i></a>, <a class="txt" href="sublemma.html#E5:70_1_2_1"><i><font color="Green" title="E71">A21</font></i></a>, <a class="ref" href="sublemma.html#T50" target="_self" title="SUBLEMMA:th.50">Th50</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:70_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:70_1_2"><i><font color="Green" title="E66">A16</font></i></a>, <a class="txt" href="sublemma.html#E3:70_1_2"><i><font color="Green" title="E67">A17</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E19:70_1"/>
 <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E17:70_1"><i><font color="Green" title="E65">A15</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" title="E57">A22</font></i>: 
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span>
 ;<br/><a NAME="E2:70_2"/><i><font color="Green" title="E58">A23</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/><a NAME="E3:70_2"/><i><font color="Green" title="E59">A24</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T45" title="CQC_LANG:th.45">CQC_LANG:45</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">F</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><b>thus </b><a NAME="E5:70_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:70_2_1_1"/><i><font color="Green" title="E60">A25</font></i>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:70_2_1_1"/><i><font color="Green" title="E61">A26</font></i>: 
( <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="sublemma.html#E4:70_2_1_1"><i><font color="Green" title="E60">A25</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>set </b><font color="Maroon" title="c12">q</font> =  <a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c13">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> ;<br/><a NAME="E6:70_2_1_1"/>
 <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">xSQ</font> <a href="substut1.html#K6" title="SUBSTUT1:func.6">|</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D6" title="SUBSTUT1:def.6">SUBSTUT1:def 6</a></i>;<br/><a NAME="E7:70_2_1_1"/><b>then </b>
 <a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E8:70_2_1_1"/><b>then </b>
 <a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E9:70_2_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E10:70_2_1_1"/><b>then </b><i><font color="Green" title="E62">A27</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:70_2_1_1"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E5:70_2_1_1"><i><font color="Green" title="E61">A26</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E12:70_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y1</font> where <font color="Olive" title="b1">y1</font> is   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> : ( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive" title="b1">y1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y1</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="sublemma.html#E10:70_2_1_1"><i><font color="Green" title="E62">A27</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E14:70_2_1_1"/><i><font color="Green" title="E63">A28</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span> &amp; <font color="Maroon" title="c14">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">xSQ</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">xSQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 ;<br/><b>thus </b><a NAME="E15:70_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sublemma.html#E5:70_2_1_1"><i><font color="Green" title="E61">A26</font></i></a>, <a class="txt" href="sublemma.html#E14:70_2_1_1"><i><font color="Green" title="E63">A28</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:70_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:70_2"><i><font color="Green" title="E58">A23</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E7:70_2"/><b>then </b><i><font color="Green" title="E60">A29</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T32" title="FUNCT_4:th.32">FUNCT_4:32</a></i>;<br/><a NAME="E8:70_2"/>
<font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,<font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D1" title="VALUAT_1:def.1">VALUAT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:70_2"/><i><font color="Green" title="E61">A30</font></i>: 
( <font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p4">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p5"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><a NAME="E11:70_2"/><i><font color="Green" title="E62">A31</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E10:70_2"><i><font color="Green" title="E61">A30</font></i></a></i>;<br/><a NAME="E12:70_2"/><i><font color="Green" title="E63">A32</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/><a NAME="E13:70_2"/><i><font color="Green" title="E64">A33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c10">F</font></span>)</span> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E10:70_2"><i><font color="Green" title="E61">A30</font></i></a>, <a class="txt" href="sublemma.html#E11:70_2"><i><font color="Green" title="E62">A31</font></i></a>, <a class="txt" href="sublemma.html#E12:70_2"><i><font color="Green" title="E63">A32</font></i></a>, <a class="ref" href="funct_7.html#T10" title="FUNCT_7:th.10">FUNCT_7:10</a></i>;<br/><a NAME="E14:70_2"/>
<font color="Maroon" title="c4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E2:70"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="sublemma.html#E1:70_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="substut1.html#D13" title="SUBSTUT1:def.13">SUBSTUT1:def 13</a></i>;<br/><a NAME="E15:70_2"/><b>then </b>
 <a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">F</font>
 <b>by </b><i><a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E16:70_2"/><b>then </b><i><font color="Green" title="E65">A34</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E7:70_2"><i><font color="Green" title="E60">A29</font></i></a>, <a class="txt" href="sublemma.html#E13:70_2"><i><font color="Green" title="E64">A33</font></i></a>, <a class="ref" href="substut1.html#D2" title="SUBSTUT1:def.2">SUBSTUT1:def 2</a></i>;<br/><a NAME="E17:70_2"/>
<font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">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" title="E66">A35</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E2:70_2_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">F</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E10:70_2"><i><font color="Green" title="E61">A30</font></i></a>, <a class="txt" href="sublemma.html#E12:70_2"><i><font color="Green" title="E63">A32</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E3:70_2_2"/><b>then </b><i><font color="Green" title="E67">A36</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T23" title="RELAT_1:th.23">RELAT_1:23</a></i>;<br/>
<a NAME="E4:70_2_2"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c12">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:70_2_2_1"/><i><font color="Green" title="E68">A37</font></i>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span>
 ;<br/>

<a NAME="E2:70_2_2_1"/><i><font color="Green" title="E69">A38</font></i>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:70_2_2"><i><font color="Green" title="E66">A35</font></i></a>, <a class="txt" href="sublemma.html#E1:70_2_2_1"><i><font color="Green" title="E68">A37</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:70_2_2_1"/><b>then </b><i><font color="Green" title="E70">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>;<br/>
<a NAME="E4:70_2_2_1"/>
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E3:70_2_2"><i><font color="Green" title="E67">A36</font></i></a>, <a class="txt" href="sublemma.html#E1:70_2_2_1"><i><font color="Green" title="E68">A37</font></i></a>, <a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/>
<a NAME="E5:70_2_2_1"/><b>then </b><i><font color="Green" title="E71">A40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E6:70_2_2_1"/>
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E3:70_2"><i><font color="Green" title="E59">A24</font></i></a>, <a class="txt" href="sublemma.html#E2:70_2_2_1"><i><font color="Green" title="E69">A38</font></i></a>, <a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>;<br/>
<a NAME="E7:70_2_2_1"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p1">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p2">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:70"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="sublemma.html#E1:70_2"><i><font color="Green" title="E57">A22</font></i></a>, <a class="ref" href="sublemma.html#T53" target="_self" title="SUBLEMMA:th.53">Th53</a></i>;<br/>
<b>hence </b><a NAME="E8:70_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:70_2_2_1"><i><font color="Green" title="E70">A39</font></i></a>, <a class="txt" href="sublemma.html#E5:70_2_2_1"><i><font color="Green" title="E71">A40</font></i></a>, <a class="ref" href="sublemma.html#T50" target="_self" title="SUBLEMMA:th.50">Th50</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:70_2_2"/>
<font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">F</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:70_2_2"><i><font color="Green" title="E66">A35</font></i></a>, <a class="txt" href="sublemma.html#E3:70_2_2"><i><font color="Green" title="E67">A36</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:70_2"/>
 <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E16:70_2"><i><font color="Green" title="E65">A34</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:70"/>
(  <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p4">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p5">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p0"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" title="SUBLEMMA:func.14">NEx_Val</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p5">(<span class="default"><a href="substut1.html#K32" title="SUBSTUT1:func.32">@</a> <span class="p0">(<span class="default"><a href="sublemma.html#K10" title="SUBLEMMA:func.10">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" title="SUBLEMMA:func.8">[</a><span class="default"><font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font></span><a href="sublemma.html#K8" title="SUBLEMMA:func.8">]</a></span>,<font color="Maroon" title="c5">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c8">a</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span></span>)</span>,<font color="Maroon" title="c5">xSQ</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="sublemma.html#E3:70"><i><font color="Green" title="E56">A3</font></i></a></i>;<br/>


</div>
