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

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>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">J</font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:97"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">v</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font><br/> for <font color="Olive">vS</font>, <font color="Olive">vS1</font>, <font color="Olive">vS2</font> being  <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>  st ( 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="Olive">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> <font color="Maroon">p</font> ) &amp; ( 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="Olive">vS2</font> holds <br/><font color="Olive">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> ) &amp;  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS2</font> holds <br/>( <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Olive">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> iff <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Olive">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Olive">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 ;<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">vS</font> be   <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">vS1</font> be   <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">vS2</font> be   <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E2:97"/><i><font color="Green">E56</font></i>: 
( ( 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>,<font color="Maroon">p</font></span>)</span> ) &amp; ( 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> ) &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font> )
 ;<br/>

<b>set </b><font color="Maroon">vS1'</font> = <font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">vS2'</font> = <font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>
<a NAME="E3:97"/><i><font color="Green">E57</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<a NAME="E4:97"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E5:97"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E6:97"/><i><font color="Green">E59</font></i>: 
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> <span class="p1">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> 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> <font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="sublemma.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E7:97"/><i><font color="Green">E60</font></i>: 
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> <span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>
 
<b>by </b><i>, , <a class="ref" href="sublemma.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E8:97"/><i><font color="Green">E61</font></i>: 
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) iff for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:97_1"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) implies for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E2:97_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="sublemma.html#E5:97"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E3:97_1_1"/><b>then </b>
( <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> iff <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:97_1_1"/>
<font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T9" target="_self">Th9</a></i>;<br/>


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

<b>thus </b><a NAME="E2:97_1"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) implies for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_1_2"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E2:97_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="sublemma.html#E5:97"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E3:97_1_2"/><b>then </b>
( <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> iff <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:97_1_2"/>
<font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T9" target="_self">Th9</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E9:97"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:97_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="sublemma.html#E5:97"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E2:97_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</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"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T36">FUNCT_4:36</a></i>;<br/>
<a NAME="E3:97_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/>
<a NAME="E4:97_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="sublemma.html#E5:97"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E5:97_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></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"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_4.html#T36">FUNCT_4:36</a></i>;<br/>
<a NAME="E6:97_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</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"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font>
 ;<br/><a NAME="E2:97_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vS1</font>
 <b>by </b><i/>;<br/><div><i><font color="Green">E129</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font>
 ;<br/><a NAME="E2:97_2_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vS2</font>
 <b>by </b><i/>;<br/><a NAME="E3:97_2_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E4:97_2_1_1"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="sublemma.html#E5:97"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E5:97_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sublemma.html#T13" target="_self">Th13</a>, <a class="ref" href="sublemma.html#T15" target="_self">Th15</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font>
 ;<br/><a NAME="E2:97_2_1_2"/><b>then </b>
<font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E3:97_2_1_2"/><b>then </b>
<font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vS2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E4:97_2_1_2"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">vS2</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>;<br/><a NAME="E5:97_2_1_2"/><i><font color="Green">E148</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E6:97_2_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E7:97_2_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E5:97"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E8:97_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sublemma.html#T13" target="_self">Th13</a>, <a class="ref" href="sublemma.html#T15" target="_self">Th15</a>, <a class="ref" href="sublemma.html#T16" target="_self">Th16</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:97_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font>
 ;<br/><a NAME="E2:97_2_2"/><b>then </b>
<font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><a NAME="E3:97_2_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vS1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E4:97_2_2"/><b>then </b><i><font color="Green">E150</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">vS1</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>;<br/><a NAME="E5:97_2_2"/><i><font color="Green">E151</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><div><i><font color="Green">E152</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font>
 ;<br/><a NAME="E2:97_2_2_1"/><b>then </b><i><font color="Green">E153</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vS2</font>
 <b>by </b><i/>;<br/><a NAME="E3:97_2_2_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E4:97_2_2_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E5:97"><i><font color="Green">E58</font></i></a></i>;<br/><b>hence </b><a NAME="E5:97_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sublemma.html#T14" target="_self">Th14</a>, <a class="ref" href="sublemma.html#T15" target="_self">Th15</a>, <a class="ref" href="sublemma.html#T17" target="_self">Th17</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:97_2_2_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font>
 ;<br/><a NAME="E2:97_2_2_2"/><b>then </b>
<font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><b>hence </b><a NAME="E3:97_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="sublemma.html#T13" target="_self">Th13</a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:97_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:97_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS1</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/>
<a NAME="E10:97_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></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"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS</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> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/>
<a NAME="E11:97_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/>
<a NAME="E12:97_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/>
<a NAME="E13:97_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E14:97_2"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:97"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) iff for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:97_3"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) implies for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_3_1"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E2:97_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:97_3_1"/>
<font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a></i>;<br/>


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

<b>thus </b><a NAME="E2:97_3"/>
( ( for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) implies for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:97_3_2"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E2:97_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">vS</font> <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></span>)</span> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:97_3_2"/>
<font color="Maroon">J</font>,<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font></span>)</span> <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"><font color="Maroon">x</font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p3">(<span class="default"><font color="Maroon">vS1</font> <a href="relset_1.html#K8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p5"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">vS2</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T11" target="_self">Th11</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:97"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> iff <font color="Maroon">J</font>,<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"><font color="Maroon">vS</font> <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> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/>


</div>
