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

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c2">p</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>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> <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default">$1 <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> $1;<br/>
<a NAME="E1:54"/><i><font color="Green" title="E32">A1</font></i>: 
for <font color="Olive" title="b1">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> <br/> for <font color="Olive" title="b2">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font><br/> for <font color="Olive" title="b3">l</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Olive" title="b1">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b3">l</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c3">k</font>;<br/><b>let </b><font color="Maroon" title="c5">l</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c3">k</font>;<br/>





<b>set </b><font color="Maroon" title="c6">F1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <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> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">ll</font> =  <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">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="c1">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">F2</font> = <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="c5">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="c1">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="c5">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="c1">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="c5">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="c1">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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span> ;<br/>
<a NAME="E1:54_1"/><i><font color="Green" title="E33">A2</font></i>: 
(  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c5">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <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> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  &amp;  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">l</font>,<span class="p3">(<span class="default"><span class="p4">(<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="c1">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">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="c1">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="c5">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="c1">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="c5">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="c1">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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T77" title="QC_LANG3:th.77">QC_LANG3:77</a></i>;<br/>
<a NAME="E2:54_1"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">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="c1">x</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#D3" target="_self" title="CQC_LANG:def.3">Def3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:54_1_1"/>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <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> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  implies <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c5">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="c1">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="c5">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="c1">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="c5">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="c1">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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <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> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E3:54_1_1_1"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>and </b><br/><a NAME="E4:54_1_1_1"/><i><font color="Green" title="E36">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">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="c5">l</font> )
 <b>and </b><br/><a NAME="E5:54_1_1_1"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</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> 
 ;<br/>
<a NAME="E6:54_1_1_1"/>
<font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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> 
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E5:54_1_1_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="qc_lang3.html#T40" title="QC_LANG3:th.40">QC_LANG3:40</a></i>;<br/>
<a NAME="E7:54_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">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="c1">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E4:54_1_1_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="cqc_lang.html#T11" target="_self" title="CQC_LANG:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E8:54_1_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c5">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="c1">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="c5">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="c1">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="c5">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="c1">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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="cqc_lang.html#E2:54_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cqc_lang.html#E3:54_1_1_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="cqc_lang.html#E4:54_1_1_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="cqc_lang.html#E5:54_1_1_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:54_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c5">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="c1">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="c5">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="c1">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="c5">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="c1">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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E4:54_1_1"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">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="c1">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>and </b><br/><a NAME="E5:54_1_1"/><i><font color="Green" title="E36">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">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="c5">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="c1">x</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E6:54_1_1"/><i><font color="Green" title="E37">A9</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">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="c1">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</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> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1_1_2"/>
<font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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="E2:54_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">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="c1">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="cqc_lang.html#E2:54_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cqc_lang.html#E5:54_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="cqc_lang.html#T11" target="_self" title="CQC_LANG:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E3:54_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="cqc_lang.html#E6:54_1_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="txt" href="cqc_lang.html#E7"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:54_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K2" title="CQC_LANG:func.2">Subst</a> <font color="Maroon" title="c5">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="c1">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="cqc_lang.html#E2:54_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cqc_lang.html#E5:54_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="cqc_lang.html#T11" target="_self" title="CQC_LANG:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E9:54_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <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> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="cqc_lang.html#E2:54_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cqc_lang.html#E4:54_1_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="cqc_lang.html#E5:54_1_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="cqc_lang.html#E6:54_1_1"><i><font color="Green" title="E37">A9</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:54_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">l</font> <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> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c5">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="c1">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="c5">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="c1">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="c5">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="c1">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#K3" title="QC_LANG1:func.3">fixed_QC-variables</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E5:54_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c5">l</font>]
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:54_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="cqc_lang.html#T30" target="_self" title="CQC_LANG:th.30">Th30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:54"/><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ]
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T28" target="_self" title="CQC_LANG:th.28">Th28</a></i>;<br/>
<a NAME="E3:54"/><i><font color="Green" title="E34">A11</font></i>: 
for <font color="Olive" title="b1">p</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Olive" title="b1">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</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>assume </b><a NAME="E1:54_2"/><i><font color="Green" title="E35">A12</font></i>: 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c3">p</font>
 ;<br/>

<b>thus </b> <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T32" target="_self" title="CQC_LANG:th.32">Th32</a></i>
<br/>.= 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:54_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="qc_lang3.html#T78" title="QC_LANG3:th.78">QC_LANG3:78</a></i>
<br/>.= 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c3">p</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T78" title="QC_LANG3:th.78">QC_LANG3:78</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:54"/><i><font color="Green" title="E35">A13</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Olive" title="b2">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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>assume </b><a NAME="E1:54_3"/><i><font color="Green" title="E36">A14</font></i>: 
(  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c3">p</font> &amp;  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c4">q</font> )
 ;<br/>

<b>thus </b> <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T34" target="_self" title="CQC_LANG:th.34">Th34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c4">q</font></span>)</span>
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:54_3"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="qc_lang3.html#T80" title="QC_LANG3:th.80">QC_LANG3:80</a></i>
<br/>.= 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c4">q</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T80" title="QC_LANG3:th.80">QC_LANG3:80</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:54"/><i><font color="Green" title="E36">A15</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><br/> for <font color="Olive" title="b2">p</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><b>proof </b></a><div class="add">

<b>let </b><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> 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>assume </b><a NAME="E1:54_4"/><i><font color="Green" title="E37">A16</font></i>: 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c4">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_4_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 ;<br/><b>hence </b> <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T38" target="_self" title="CQC_LANG:th.38">Th38</a></i>
<br/>.= 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c4">p</font>
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:54_4"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="qc_lang3.html#T81" title="QC_LANG3:th.81">QC_LANG3:81</a></i>
<br/>.= 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T81" title="QC_LANG3:th.81">QC_LANG3:81</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:54_4"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">p</font>]
 <b>by </b><i><a class="ref" href="cqc_lang.html#T37" target="_self" title="CQC_LANG:th.37">Th37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:54"/>
for <font color="Olive" title="b1">p</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">p</font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S1" title="QC_LANG1:sch.1">QC_LANG1:sch 1</a>(<a class="txt" href="cqc_lang.html#E1:54"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="cqc_lang.html#E2:54"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="cqc_lang.html#E3:54"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="cqc_lang.html#E4:54"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="cqc_lang.html#E5:54"><i><font color="Green" title="E36">A15</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:54"/>
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c2">p</font>
 ;<br/>


</div>
