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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c6">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c7">s'</font> be   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>;<br/>













<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ] <b>means </b>for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>;<br/>
<div><i><font color="Green" title="E28">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ;<br/><b>thus </b><a NAME="E1:40_1"/>
( <font color="Maroon" title="c8">s</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c8">s</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>
 ;<br/>

<b>thus </b><a NAME="E2:40_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c13">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:40_1_1_1"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c14">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c15">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c14">k</font>, <font color="Maroon" title="c16">l</font> being    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c14">k</font><b> such that </b><br/><a NAME="E4:40_1_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c16">l</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="qc_lang1.html#D17" title="QC_LANG1:def.17">QC_LANG1:def 17</a></i>;<br/>
<a NAME="E5:40_1_1_1"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="valuat_1.html#E4:40_1_1_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="cqc_lang.html#T30" title="CQC_LANG:th.30">CQC_LANG:30</a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">lx</font> =  <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c18">ly</font> =  <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span>;<br/>
<a NAME="E6:40_1_1_1"/><i><font color="Green" title="E34">A7</font></i>: 
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K4" title="QC_LANG1:func.4">free_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E5:40_1_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="cqc_lang.html#T17" title="CQC_LANG:th.17">CQC_LANG:17</a></i>;<br/>
<a NAME="E7:40_1_1_1"/>
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K4" title="QC_LANG1:func.4">free_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E5:40_1_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="cqc_lang.html#T17" title="CQC_LANG:th.17">CQC_LANG:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">lx</font> =  <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c9">x</font></span>)</span>, <font color="Maroon" title="c20">ly</font> =  <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c16">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c10">y</font></span>)</span> as   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c14">k</font> <b>by </b><i><a class="txt" href="valuat_1.html#E6:40_1_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="cqc_lang.html#T15" title="CQC_LANG:th.15">CQC_LANG:15</a></i>;<br/>
<a NAME="E9:40_1_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="valuat_1.html#D8" target="_self" title="VALUAT_1:def.8">Def8</a></i>;<br/>
<a NAME="E10:40_1_1_1"/><i><font color="Green" title="E36">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">ly</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="valuat_1.html#D8" target="_self" title="VALUAT_1:def.8">Def8</a></i>;<br/>
<a NAME="E11:40_1_1_1"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:40_1_1_1_1_1"/><i><font color="Green" title="E38">A11</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> )
 ;<br/><a NAME="E2:40_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E39">A12</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">l</font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E9:40_1_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="qc_lang1.html#D8" title="QC_LANG1:def.8">QC_LANG1:def 8</a></i>;<br/><a NAME="E3:40_1_1_1_1_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><i><font color="Green" title="E41">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_1_1_1_1"/>
<font color="Maroon" title="c16">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:40_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E42">A15</font></i>: 
( <font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c20">ly</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="cqc_lang.html#T11" title="CQC_LANG:th.11">CQC_LANG:11</a></i>;<br/><a NAME="E3:40_1_1_1_1_1_1"/>
( <font color="Maroon" title="c13">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c13">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">ly</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E9:40_1_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="valuat_1.html#D8" target="_self" title="VALUAT_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E4:40_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_1_1_1_1_1"><i><font color="Green" title="E42">A15</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_1_1_1_2"/>
<font color="Maroon" title="c16">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:40_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A16</font></i>: 
( <font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c20">ly</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_1_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_1_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="cqc_lang.html#T11" title="CQC_LANG:th.11">CQC_LANG:11</a></i>;<br/><a NAME="E3:40_1_1_1_1_1_2"/>
( <font color="Maroon" title="c13">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">lx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> &amp; <font color="Maroon" title="c13">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">ly</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E9:40_1_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="valuat_1.html#D8" target="_self" title="VALUAT_1:def.8">Def8</a></i>;<br/><b>hence </b><a NAME="E4:40_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_1_1_1_1_2"><i><font color="Green" title="E42">A16</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:40_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_1_1_1_1"><i><font color="Green" title="E41">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_1_1_1_1"/>
<font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c20">ly</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E9:40_1_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="valuat_1.html#E10:40_1_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E38">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/><a NAME="E2:40_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#K7" title="VALUAT_1:func.7">.</a> <font color="Maroon" title="c15">P</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:40_1_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="valuat_1.html#T16" target="_self" title="VALUAT_1:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E3:40_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:40_1_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="valuat_1.html#E11:40_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="valuat_1.html#T16" target="_self" title="VALUAT_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E2:40_1_1_1_3"/><b>then </b>
not <font color="Maroon" title="c13">v</font> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c19">lx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">J</font> <a href="valuat_1.html#K7" title="VALUAT_1:func.7">.</a> <font color="Maroon" title="c15">P</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:40_1_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="valuat_1.html#T17" target="_self" title="VALUAT_1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E3:40_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:40_1_1_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="valuat_1.html#E11:40_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="ref" href="valuat_1.html#T17" target="_self" title="VALUAT_1:th.17">Th17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:40_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E12:40_1_1_1"><i><font color="Green" title="E38">A17</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E2:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:40_1_2"/><i><font color="Green" title="E29">A18</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c13">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:40_1_2"/>
<font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ;<br/>

<a NAME="E3:40_1_2"/>
( <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_2"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="cqc_lang.html#T28" title="CQC_LANG:th.28">CQC_LANG:28</a></i>;<br/>
<b>hence </b><a NAME="E4:40_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:40_1"/>
( <font color="Maroon" title="c8">s</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_3"/><i><font color="Green" title="E29">A19</font></i>: 
<font color="Maroon" title="c8">s</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>
 <b>and </b><br/><a NAME="E2:40_1_3"/><i><font color="Green" title="E30">A20</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>
 ;<br/>

<b>thus </b><a NAME="E3:40_1_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:40_1_3_1"/><i><font color="Green" title="E31">A21</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c13">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:40_1_3_1"/><i><font color="Green" title="E32">A22</font></i>: 
<font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ;<br/>

<a NAME="E3:40_1_3_1"/><i><font color="Green" title="E33">A23</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font></span>)</span> &amp; <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_3"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="cqc_lang.html#T31" title="CQC_LANG:th.31">CQC_LANG:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">pa</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_3_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="cqc_lang.html#T18" title="CQC_LANG:th.18">CQC_LANG:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">qa</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_3_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_3_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="cqc_lang.html#T18" title="CQC_LANG:th.18">CQC_LANG:18</a></i>;<br/>
<div><i><font color="Green" title="E34">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/><a NAME="E2:40_1_3_1_1"/><b>then </b>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">pa</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_3_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_3_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="valuat_1.html#T20" target="_self" title="VALUAT_1:th.20">Th20</a></i>;<br/><a NAME="E3:40_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">pa</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41" title="MARGREL1:th.41">MARGREL1:41</a></i>;<br/><a NAME="E4:40_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c15">qa</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_3"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_3_1"><i><font color="Green" title="E32">A22</font></i></a></i>;<br/><a NAME="E5:40_1_3_1_1"/><b>then </b>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c15">qa</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41" title="MARGREL1:th.41">MARGREL1:41</a></i>;<br/><b>hence </b><a NAME="E6:40_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_3_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_3_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="valuat_1.html#T20" target="_self" title="VALUAT_1:th.20">Th20</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E2:40_1_3_1_2"/><b>then </b>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">pa</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_3_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_3_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="valuat_1.html#T20" target="_self" title="VALUAT_1:th.20">Th20</a></i>;<br/><a NAME="E3:40_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">pa</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41" title="MARGREL1:th.41">MARGREL1:41</a></i>;<br/><a NAME="E4:40_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c15">qa</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_3"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_3_1"><i><font color="Green" title="E32">A22</font></i></a></i>;<br/><a NAME="E5:40_1_3_1_2"/><b>then </b>
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c15">qa</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41" title="MARGREL1:th.41">MARGREL1:41</a></i>;<br/><b>hence </b><a NAME="E6:40_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_3_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_3_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="valuat_1.html#T20" target="_self" title="VALUAT_1:th.20">Th20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:40_1_3_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E6:40_1_3_1"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E4:40_1"/>
( <font color="Maroon" title="c8">s</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_4"/><i><font color="Green" title="E29">A25</font></i>: 
<font color="Maroon" title="c8">s</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>
 <b>and </b><br/><a NAME="E2:40_1_4"/><i><font color="Green" title="E30">A26</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>
 <b>and </b><br/><a NAME="E3:40_1_4"/><i><font color="Green" title="E31">A27</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>
 ;<br/>

<b>thus </b><a NAME="E4:40_1_4"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c11">p</font>, <font color="Maroon" title="c12">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:40_1_4_1"/><i><font color="Green" title="E32">A28</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> &amp; <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c12">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c13">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:40_1_4_1"/><i><font color="Green" title="E33">A29</font></i>: 
<font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font>
 ;<br/>

<a NAME="E3:40_1_4_1"/><i><font color="Green" title="E34">A30</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font></span>)</span> &amp; <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_4"><i><font color="Green" title="E29">A25</font></i></a>, <a class="ref" href="cqc_lang.html#T33" title="CQC_LANG:th.33">CQC_LANG:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">pla</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c15">pra</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_4_1"><i><font color="Green" title="E32">A28</font></i></a>, <a class="ref" href="cqc_lang.html#T19" title="CQC_LANG:th.19">CQC_LANG:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">qla</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font>, <font color="Maroon" title="c17">qra</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_4_1"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_4_1"><i><font color="Green" title="E34">A30</font></i></a>, <a class="ref" href="cqc_lang.html#T19" title="CQC_LANG:th.19">CQC_LANG:19</a></i>;<br/>
<div><i><font color="Green" title="E35">A31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/><a NAME="E2:40_1_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">pla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c15">pra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_4_1"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_4_1"><i><font color="Green" title="E34">A30</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self" title="VALUAT_1:th.22">Th22</a></i>;<br/><a NAME="E3:40_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">pla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c15">pra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T45" title="MARGREL1:th.45">MARGREL1:45</a></i>;<br/><a NAME="E4:40_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c16">qla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_4"><i><font color="Green" title="E30">A26</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_4"><i><font color="Green" title="E31">A27</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_4_1"><i><font color="Green" title="E33">A29</font></i></a></i>;<br/><a NAME="E5:40_1_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c16">qla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/><b>hence </b><a NAME="E6:40_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_4_1"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_4_1"><i><font color="Green" title="E34">A30</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self" title="VALUAT_1:th.22">Th22</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_4_1_2"/><i><font color="Green" title="E36">A32</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E2:40_1_4_1_2"/><b>then </b><i><font color="Green" title="E37">A33</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">pla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c15">pra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_4_1"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_4_1"><i><font color="Green" title="E34">A30</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self" title="VALUAT_1:th.22">Th22</a></i>;<br/><div><i><font color="Green" title="E38">A34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_4_1_2_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">pla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E2:40_1_4_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c16">qla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_4"><i><font color="Green" title="E30">A26</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_4_1"><i><font color="Green" title="E33">A29</font></i></a></i>;<br/><a NAME="E3:40_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#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c16">qla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45" title="MARGREL1:th.45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E4:40_1_4_1_2_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_4_1"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_4_1"><i><font color="Green" title="E34">A30</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_4_1_2"><i><font color="Green" title="E36">A32</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self" title="VALUAT_1:th.22">Th22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_4_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_4_1_2_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c15">pra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E2:40_1_4_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:40_1_4"><i><font color="Green" title="E31">A27</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_4_1"><i><font color="Green" title="E33">A29</font></i></a></i>;<br/><a NAME="E3:40_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#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c16">qla</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qra</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45" title="MARGREL1:th.45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E4:40_1_4_1_2_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_4_1"><i><font color="Green" title="E32">A28</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_4_1"><i><font color="Green" title="E34">A30</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_4_1_2"><i><font color="Green" title="E36">A32</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self" title="VALUAT_1:th.22">Th22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:40_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_4_1_2"><i><font color="Green" title="E37">A33</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_4_1_2"><i><font color="Green" title="E38">A34</font></i></a>, <a class="ref" href="margrel1.html#T45" title="MARGREL1:th.45">MARGREL1:45</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:40_1_4_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c11">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c12">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E6:40_1_4_1"><i><font color="Green" title="E35">A31</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/>


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


</div><b>end;</b></div><b>thus </b><a NAME="E5:40_1"/>
( <font color="Maroon" title="c8">s</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">s</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:40_1_5"/><i><font color="Green" title="E29">A35</font></i>: 
<font color="Maroon" title="c8">s</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>
 <b>and </b><br/><a NAME="E2:40_1_5"/><i><font color="Green" title="E30">A36</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">xx</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>, <font color="Maroon" title="c10">pp</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> <b> such that </b><br/><a NAME="E4:40_1_5"/><i><font color="Green" title="E31">A37</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c9">xx</font>,<font color="Maroon" title="c10">pp</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5"><i><font color="Green" title="E29">A35</font></i></a>, <a class="ref" href="qc_lang1.html#D20" title="QC_LANG1:def.20">QC_LANG1:def 20</a></i>;<br/>
<a NAME="E5:40_1_5"/><i><font color="Green" title="E32">A38</font></i>: 
<font color="Maroon" title="c9">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5"><i><font color="Green" title="E29">A35</font></i></a>, <a class="txt" href="valuat_1.html#E4:40_1_5"><i><font color="Green" title="E31">A37</font></i></a>, <a class="ref" href="qc_lang1.html#D26" title="QC_LANG1:def.26">QC_LANG1:def 26</a></i>;<br/>
<b>thus </b><a NAME="E6:40_1_5"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Olive" title="b2">y</font> holds <br/>for <font color="Olive" title="b5">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Olive" title="b4">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:40_1_5_1"/><i><font color="Green" title="E33">A39</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font> &amp; <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c12">y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c15">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E2:40_1_5_1"/><i><font color="Green" title="E34">A40</font></i>: 
<font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font>
 ;<br/>

<div><i><font color="Green" title="E35">A41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_5_1_1"/><i><font color="Green" title="E36">A42</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 ;<br/><a NAME="E2:40_1_5_1_1"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5"><i><font color="Green" title="E29">A35</font></i></a>, <a class="ref" href="cqc_lang.html#T36" title="CQC_LANG:th.36">CQC_LANG:36</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">ps</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5_1"><i><font color="Green" title="E33">A39</font></i></a>, <a class="ref" href="cqc_lang.html#T23" title="CQC_LANG:th.23">CQC_LANG:23</a></i>;<br/><a NAME="E4:40_1_5_1_1"/><i><font color="Green" title="E37">A43</font></i>: 
 <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<font color="Maroon" title="c16">ps</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5"><i><font color="Green" title="E29">A35</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1"><i><font color="Green" title="E33">A39</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_1"><i><font color="Green" title="E36">A42</font></i></a>, <a class="ref" href="cqc_lang.html#T36" title="CQC_LANG:th.36">CQC_LANG:36</a></i>;<br/><div><i><font color="Green" title="E38">A44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_5_1_1_1"/><i><font color="Green" title="E39">A45</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 ;<br/><a NAME="E2:40_1_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c12">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5"><i><font color="Green" title="E29">A35</font></i></a>, <a class="ref" href="cqc_lang.html#T36" title="CQC_LANG:th.36">CQC_LANG:36</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">qs</font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5_1"><i><font color="Green" title="E33">A39</font></i></a>, <a class="ref" href="cqc_lang.html#T23" title="CQC_LANG:th.23">CQC_LANG:23</a></i>;<br/><a NAME="E4:40_1_5_1_1_1"/><i><font color="Green" title="E40">A46</font></i>: 
 <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<font color="Maroon" title="c17">qs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5"><i><font color="Green" title="E29">A35</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1"><i><font color="Green" title="E33">A39</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_1_1"><i><font color="Green" title="E39">A45</font></i></a>, <a class="ref" href="cqc_lang.html#T36" title="CQC_LANG:th.36">CQC_LANG:36</a></i>;<br/><a NAME="E5:40_1_5_1_1_1"/><i><font color="Green" title="E41">A47</font></i>: 
(  <a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<font color="Maroon" title="c16">ps</font></span>)</span>,<font color="Maroon" title="c6">J</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c16">ps</font>,<font color="Maroon" title="c6">J</font></span>)</span> &amp;  <a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<font color="Maroon" title="c17">qs</font></span>)</span>,<font color="Maroon" title="c6">J</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="valuat_1.html#K3" title="VALUAT_1:func.3">FOR_ALL</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font></span>)</span>,<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font></span>)</span> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E24"><i><font color="Green" title="E8">Lm1</font></i></a></i>;<br/><div><i><font color="Green" title="E42">A48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_5_1_1_1_1"/><i><font color="Green" title="E43">A49</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 ;<br/><a NAME="E2:40_1_5_1_1_1_1"/>
for <font color="Olive" title="b1">v1</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">v1</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:40_1_5_1_1_1_1_1"/><i><font color="Green" title="E44">A50</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font>
 ;<br/>

<a NAME="E2:40_1_5_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E45">A51</font></i>: 
( <font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5_1_1"><i><font color="Green" title="E36">A42</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_1_1"><i><font color="Green" title="E39">A45</font></i></a></i>;<br/>
<a NAME="E3:40_1_5_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c16">ps</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_5_1_1"><i><font color="Green" title="E37">A43</font></i></a>, <a class="txt" href="valuat_1.html#E5:40_1_5_1_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_1_1_1"><i><font color="Green" title="E43">A49</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_1_1_1_1"><i><font color="Green" title="E44">A50</font></i></a>, <a class="ref" href="valuat_1.html#T8" target="_self" title="VALUAT_1:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:40_1_5_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_5"><i><font color="Green" title="E30">A36</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_5_1"><i><font color="Green" title="E34">A40</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_5_1_1_1_1_1"><i><font color="Green" title="E45">A51</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:40_1_5_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_5_1_1_1"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="valuat_1.html#E5:40_1_5_1_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="ref" href="valuat_1.html#T8" target="_self" title="VALUAT_1:th.8">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_5_1_1_1_2"/><i><font color="Green" title="E43">A52</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/><a NAME="E2:40_1_5_1_1_1_2"/>
 ex <font color="Olive" title="b1">v1</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font> st <br/>( <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c18">v1</font> being    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E2:40_1_5_1_1_1_2_1"/><i><font color="Green" title="E44">A53</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c16">ps</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>and </b><br/><a NAME="E3:40_1_5_1_1_1_2_1"/><i><font color="Green" title="E45">A54</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_5_1_1"><i><font color="Green" title="E37">A43</font></i></a>, <a class="txt" href="valuat_1.html#E5:40_1_5_1_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_1_1_2"><i><font color="Green" title="E43">A52</font></i></a>, <a class="ref" href="valuat_1.html#T7" target="_self" title="VALUAT_1:th.7">Th7</a></i>;<br/>
<a NAME="E4:40_1_5_1_1_1_2_1"/>
( <font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c18">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">y</font> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:40_1_5_1_1"><i><font color="Green" title="E36">A42</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_1_1"><i><font color="Green" title="E39">A45</font></i></a>, <a class="txt" href="valuat_1.html#E3:40_1_5_1_1_1_2_1"><i><font color="Green" title="E45">A54</font></i></a></i>;<br/>
<a NAME="E5:40_1_5_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="valuat_1.html#E2:40_1_5"><i><font color="Green" title="E30">A36</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_5_1"><i><font color="Green" title="E34">A40</font></i></a>, <a class="txt" href="valuat_1.html#E2:40_1_5_1_1_1_2_1"><i><font color="Green" title="E44">A53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:40_1_5_1_1_1_2_1"/>
 ex <font color="Olive" title="b1">v1</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font> st <br/>( <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c17">qs</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; ( for <font color="Olive" title="b2">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> ) )
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:40_1_5_1_1_1_2_1"><i><font color="Green" title="E45">A54</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:40_1_5_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_5_1_1_1"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="valuat_1.html#E5:40_1_5_1_1_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="ref" href="valuat_1.html#T7" target="_self" title="VALUAT_1:th.7">Th7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:40_1_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E6:40_1_5_1_1_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="xboolean.html#D3" title="XBOOLEAN:def.3">XBOOLEAN:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_5_1_1_2"/><i><font color="Green" title="E39">A55</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 ;<br/><a NAME="E2:40_1_5_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c10">pp</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_5"><i><font color="Green" title="E31">A37</font></i></a>, <a class="txt" href="valuat_1.html#E5:40_1_5"><i><font color="Green" title="E32">A38</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1"><i><font color="Green" title="E33">A39</font></i></a>, <a class="ref" href="cqc_lang.html#T37" title="CQC_LANG:th.37">CQC_LANG:37</a></i>;<br/><b>hence </b><a NAME="E3:40_1_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_5"><i><font color="Green" title="E31">A37</font></i></a>, <a class="txt" href="valuat_1.html#E5:40_1_5"><i><font color="Green" title="E32">A38</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1"><i><font color="Green" title="E33">A39</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_1_2"><i><font color="Green" title="E39">A55</font></i></a>, <a class="ref" href="cqc_lang.html#T40" title="CQC_LANG:th.40">CQC_LANG:40</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:40_1_5_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:40_1_5_1_1"><i><font color="Green" title="E38">A44</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_5_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_5_1_2"/><i><font color="Green" title="E36">A56</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c8">s</font>
 ;<br/><a NAME="E2:40_1_5_1_2"/><b>then </b>
<font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">pp</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_5"><i><font color="Green" title="E31">A37</font></i></a>, <a class="txt" href="valuat_1.html#E5:40_1_5"><i><font color="Green" title="E32">A38</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1"><i><font color="Green" title="E33">A39</font></i></a>, <a class="ref" href="cqc_lang.html#T37" title="CQC_LANG:th.37">CQC_LANG:37</a></i>;<br/><b>hence </b><a NAME="E3:40_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:40_1_5"><i><font color="Green" title="E31">A37</font></i></a>, <a class="txt" href="valuat_1.html#E5:40_1_5"><i><font color="Green" title="E32">A38</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1"><i><font color="Green" title="E33">A39</font></i></a>, <a class="txt" href="valuat_1.html#E1:40_1_5_1_2"><i><font color="Green" title="E36">A56</font></i></a>, <a class="ref" href="cqc_lang.html#T40" title="CQC_LANG:th.40">CQC_LANG:40</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:40_1_5_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c14">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">v</font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:40_1_5_1"><i><font color="Green" title="E35">A41</font></i></a></i>;<br/>


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


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:40"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S2" title="QC_LANG1:sch.2">QC_LANG1:sch 2</a>(<a class="txt" href="valuat_1.html#E1:40"><i><font color="Green" title="E28">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:40"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s'</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s'</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c3">y</font> implies for <font color="Olive" title="b1">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> )
 ;<br/>


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