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

<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">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><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">J</font> be    <a href="valuat_1.html#M1" target="_self">interpretation</a> of <font color="Maroon">A</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ] <b>means </b>( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">a<sub>1</sub></font> implies for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> );<br/>
<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</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">t</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">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">ll</font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><a NAME="E1:43_1"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ll</font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <b>by </b><i><a class="ref" href="cqc_lang.html#D5">CQC_LANG:def 5</a></i>;<br/><b>thus </b><a NAME="E2:43_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9">VERUM</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <a href="cqc_lang.html#K9">VERUM</a> 
 ;<br/>

<b>thus </b><a NAME="E2:43_1_1"/>
for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">w</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:43_1_1_1"/>
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Maroon">w</font> <a href="funct_2.html#K3">.</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>
 ;<br/>

<a NAME="E2:43_1_1_1"/>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:43_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 ;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E3:43_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_2"/><i><font color="Green">E32</font></i>: 
not <font color="Maroon">x</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"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:43_1_2"/>
for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">w</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:43_1_2_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Maroon">w</font> <a href="funct_2.html#K3">.</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>
 ;<br/>

<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_2_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:43_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="valuat_1.html#K9" target="_self">'in'</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:43_1_2_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><a NAME="E4:43_1_2_1_1"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:43_1_2_1_1_1"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> )
 ;<br/><a NAME="E2:43_1_2_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:43_1_2_1_1"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E3:43_1_2_1_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:43_1_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/><a NAME="E4:43_1_2_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_2_1_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ll</font>
 
<b>by </b><i><a class="txt" href="valuat_1.html#E3:43_1_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E1:43_1_2_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E2:43_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ll</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_2_1_1_1_1"/>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:43_1_2_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">M</font> =  <a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:43_1_2_1_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:43_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="qc_lang3.html#T9">QC_LANG3:9</a></i>;<br/>
<a NAME="E3:43_1_2_1_1_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang3.html#K1">variables_in</a> <font color="Maroon">ll</font>,<a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T6">QC_LANG3:6</a></i>;<br/>
<a NAME="E4:43_1_2_1_1_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i2</font></span>)</span> where <font color="Olive">i2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i2</font> &amp; <font color="Olive">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i2</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="qc_lang3.html#D2">QC_LANG3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:43_1_2_1_1_1_2"/>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:43_1_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E1:43_1_2_1_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> <b>by </b><i><a class="txt" href="valuat_1.html#E2:43_1_2_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E7:43_1_2_1_1_1"/>
<font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:43_1_2_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="valuat_1.html#E3:43_1_2_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E8:43_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:43_1_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E3:43_1_2_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E4:43_1_2_1_1"><i><font color="Green">E39</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E6:43_1_2_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:43_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E1:43_1_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><a NAME="E7:43_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="valuat_1.html#K9" target="_self">'in'</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:43_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_2_1_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:43_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="valuat_1.html#K9" target="_self">'in'</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:43_1_2_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
not <font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><a NAME="E4:43_1_2_1_2"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:43_1_2_1_2_1"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> )
 ;<br/><a NAME="E2:43_1_2_1_2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:43_1_2_1_2_1"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font>
 <b>by </b><i>, <a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/><a NAME="E4:43_1_2_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_2_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ll</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E2:43_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ll</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_2_1_2_1_1"/>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:43_1_2_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1_2_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">M</font> =  <a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">ll</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:43_1_2_1_2_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:43_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="qc_lang3.html#T9">QC_LANG3:9</a></i>;<br/>
<a NAME="E3:43_1_2_1_2_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang3.html#K1">variables_in</a> <font color="Maroon">ll</font>,<a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T6">QC_LANG3:6</a></i>;<br/>
<a NAME="E4:43_1_2_1_2_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i2</font></span>)</span> where <font color="Olive">i2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i2</font> &amp; <font color="Olive">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i2</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="qc_lang3.html#D2">QC_LANG3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:43_1_2_1_2_1_2"/>
<font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> <b>by </b><i/>;<br/><a NAME="E7:43_1_2_1_2_1"/>
<font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:43_1_2_1"><i><font color="Green">E33</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:43_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><a NAME="E6:43_1_2_1_2"/><b>then </b>
not <font color="Maroon">w</font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:43_1_2_1"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><a NAME="E7:43_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">ll</font> <a href="valuat_1.html#K9" target="_self">'in'</a> <span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:43_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1_2_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E4:43_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:43_1_3"/><i><font color="Green">E119</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> implies for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> )
 <b>and </b><br/><a NAME="E2:43_1_3"/><i><font color="Green">E120</font></i>: 
not <font color="Maroon">x</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#K10">'not'</a> <font color="Maroon">s</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E3:43_1_3"/>
for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">w</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:43_1_3_1"/><i><font color="Green">E121</font></i>: 
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Maroon">w</font> <a href="funct_2.html#K3">.</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>
 ;<br/>

<div><i><font color="Green">E122</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_3_1_1"/><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:43_1_3_1_1"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:43_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><a NAME="E4:43_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, , , <a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/><a NAME="E5:43_1_3_1_1"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><b>hence </b><a NAME="E6:43_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_3_1_2"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:43_1_3_1_2"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:43_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><a NAME="E4:43_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, , , <a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/><a NAME="E5:43_1_3_1_2"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><b>hence </b><a NAME="E6:43_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1_3_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E5:43_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">t</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:43_1_4"/><i><font color="Green">E125</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> implies for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> )
 <b>and </b><br/><a NAME="E2:43_1_4"/><i><font color="Green">E126</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">t</font> implies for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> )
 <b>and </b><br/><a NAME="E3:43_1_4"/><i><font color="Green">E127</font></i>: 
not <font color="Maroon">x</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"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>
 ;<br/>

<a NAME="E4:43_1_4"/><i><font color="Green">E128</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>;<br/>
<b>thus </b><a NAME="E5:43_1_4"/>
for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">w</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:43_1_4_1"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Maroon">w</font> <a href="funct_2.html#K3">.</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>
 ;<br/>

<div><i><font color="Green">E130</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_4_1_1"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:43_1_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:43_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><a NAME="E4:43_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:43_1_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>hence </b><a NAME="E6:43_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_4_1_2"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:43_1_4_1_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><div><i><font color="Green">E134</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_4_1_2_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:43_1_4_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:43_1_4_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E4:43_1_4_1_2_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_4_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_4_1_2_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:43_1_4_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:43_1_4_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">t</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E4:43_1_4_1_2_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:43_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, , <a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1_4_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E6:43_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:43_1_5"/><i><font color="Green">E135</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> implies for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> )
 <b>and </b><br/><a NAME="E2:43_1_5"/><i><font color="Green">E136</font></i>: 
not <font color="Maroon">x</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">y</font>,<font color="Maroon">s</font></span>)</span>
 ;<br/>

<a NAME="E3:43_1_5"/><i><font color="Green">E137</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/>
<b>thus </b><a NAME="E4:43_1_5"/>
for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>  st ( for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">w</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:43_1_5_1"/><i><font color="Green">E138</font></i>: 
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 ;<br/>

<div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_5_1_1"/><i><font color="Green">E140</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:43_1_5_1_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K7" target="_self">FOR_ALL</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:43_1_5_1_1"/>
for <font color="Olive">v1</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>  st ( for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Olive">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v1</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:43_1_5_1_1_1"/><i><font color="Green">E142</font></i>: 
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 ;<br/>

<div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_5_1_1_1_1"/><i><font color="Green">E144</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font>
 ;<br/><b>consider </b><font color="Maroon">v2</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:43_1_5_1_1_1_1"/><i><font color="Green">E145</font></i>: 
( ( for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> ) &amp; <font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>from </b><i><a class="ref" href="valuat_1.html#S1" target="_self">VALUAT_1:sch 1</a>();<br/></i><a NAME="E4:43_1_5_1_1_1_1"/><i><font color="Green">E146</font></i>: 
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_1_5_1_1_1_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_5_1_1_1_1_1_1"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><b>hence </b><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:43_1_5_1_1_1_1_1"/>
<font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:43_1_5_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:43_1_5_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, , <a class="ref" href="valuat_1.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_5_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:43_1_5_1_1_1_2"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:43_1_5_1_1_1_2"/>
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_1_5_1_1_1_2_1"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<b>hence </b><font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:43_1_5_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:43_1_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K7" target="_self">FOR_ALL</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:43_1_5_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_5_1_2"/><i><font color="Green">E151</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:43_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K7" target="_self">FOR_ALL</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">v1</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:43_1_5_1_2"/><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>and </b><br/><a NAME="E5:43_1_5_1_2"/><i><font color="Green">E153</font></i>: 
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 <b>by </b><i/>;<br/><a NAME="E6:43_1_5_1_2"/>
 ex <font color="Olive">v2</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> st <br/>( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; ( for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Olive">v2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_2_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E154</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_5_1_2_1_1"/><i><font color="Green">E155</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font>
 ;<br/><b>consider </b><font color="Maroon">v2</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:43_1_5_1_2_1_1"/><i><font color="Green">E156</font></i>: 
( ( for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> ) &amp; <font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>from </b><i><a class="ref" href="valuat_1.html#S1" target="_self">VALUAT_1:sch 1</a>();<br/></i><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v2</font>;<br/><a NAME="E4:43_1_5_1_2_1_1"/>
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_2_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_1_5_1_2_1_1_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_5_1_2_1_1_1_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><b>hence </b><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="valuat_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:43_1_5_1_2_1_1_1"/>
<font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:43_1_5_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E6:43_1_5_1_2_1_1"/>
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_5_1_2_1_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>take </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font>;<br/><b>thus </b><a NAME="E2:43_1_5_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:43_1_5_1_2_1_2"/>
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_5_1_2_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_1_5_1_2_1_2_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E2:43_1_5_1_2_1_2_1"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">v1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> = 
<font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="valuat_1.html#D4" target="_self">Def4</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:43_1_5_1_2_1_2"/>
for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Maroon">v2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:43_1_5_1_2_1"/>
 ex <font color="Olive">v2</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font> st <br/>( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v2</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; ( for <font color="Olive">z</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">z</font> holds <br/><font color="Olive">v2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="funct_2.html#K3">.</a> <font color="Olive">z</font> ) )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:43_1_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K7" target="_self">FOR_ALL</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">s</font>,<font color="Maroon">J</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:43_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_1_5_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">y</font>,<font color="Maroon">s</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>


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


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:43"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>]
 
<b>from </b><i><a class="ref" href="cqc_lang.html#S1">CQC_LANG:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E3:43"/>
( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> implies for <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <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="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/><font color="Olive">w</font> <a href="funct_2.html#K3">.</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> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> )
 ;<br/>


</div>
