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<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">xSQ</font> be    <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>;<br/>









<b>set </b><font color="Maroon">S1</font> =  <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font>;<br/>
<b>assume </b><a NAME="E1:105"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span> is <a href="substut1.html#V3">quantifiable</a>
 ;<br/>

<a NAME="E2:105"/><i><font color="Green">E56</font></i>: 
 <a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p4">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p5">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E3:105"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
;<br/>
<a NAME="E4:105"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E6:105"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E5:105"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
;<br/>
<a NAME="E6:105"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E6:105"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E7:105"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E8:105"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E9:105"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="sublemma.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:105"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
;<br/>
<a NAME="E11:105"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E6:105"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E12:105"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p4">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p5">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p0">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="sublemma.html#T15" target="_self">Th15</a>, <a class="ref" href="funct_7.html#T10">FUNCT_7:10</a></i>;<br/>
<a NAME="E13:105"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T12" target="_self">Th12</a>, <a class="ref" href="sublemma.html#T13" target="_self">Th13</a>, <a class="ref" href="funct_7.html#T10">FUNCT_7:10</a></i>;<br/>
<b>take </b><font color="Maroon">vS1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font>;<br/>

<b>take </b><font color="Maroon">vS2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font>;<br/>

<b>thus </b><a NAME="E14:105"/>
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>

<b>assume </b><a NAME="E1:105_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font>
 ;<br/>

<a NAME="E2:105_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T13" target="_self">Th13</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E3:105_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:105_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K16" target="_self">RSub1</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E6:105_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; not <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:105_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#T19" target="_self">Th19</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E15:105"/>
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font> holds <br/><font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>

<b>assume </b><a NAME="E1:105_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font>
 ;<br/>

<a NAME="E2:105_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T16" target="_self">Th16</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E3:105_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:105_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E6:105_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:105_2"/>
<font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p5">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T19" target="_self">Th19</a>, <a class="ref" href="sublemma.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E8:105_2"/>
<font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E16:105"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> ;<br/>
<a NAME="E1:105_3"/>
 <a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xSQ</font> <a href="substut1.html#K6">|</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="substut1.html#D6">SUBSTUT1:def 6</a></i>;<br/>
<a NAME="E2:105_3"/><b>then </b>
 <a href="substut1.html#K7">RestrictSub</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E3:105_3"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E4:105_3"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E5:105_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T16" target="_self">Th16</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E6:105_3"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:105_3_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:105_3_1"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:105_3_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="qc_lang1.html#NM3">bound_QC-variable</a> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="sublemma.html#T19" target="_self">Th19</a>, <a class="ref" href="sublemma.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E6:105_3_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Maroon">y</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xSQ</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xSQ</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/><a NAME="E7:105_3_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="sublemma.html#K17" target="_self">RSub2</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span>,<font color="Maroon">xSQ</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T20" target="_self">Th20</a>, <a class="ref" href="sublemma.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">y1</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E9:105_3_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></span>)</span> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <font color="Maroon">xSQ</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E10:105_3_1"/>
contradiction
 <b>by </b><i><a class="ref" href="sublemma.html#T21" target="_self">Th21</a>, <a class="ref" href="sublemma.html#T22" target="_self">Th22</a>, <a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:105_3"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E17:105"/>
<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">S</font>,<font color="Maroon">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">xSQ</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">v</font>,<font color="Maroon">S</font>,<font color="Maroon">x</font>,<font color="Maroon">xSQ</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#T17" target="_self">Th17</a>, <a class="ref" href="sublemma.html#T18" target="_self">Th18</a></i>;<br/>


</div>
