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

<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ] <b>means </b> <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:59"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font><br/> for <font color="Olive">l</font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Olive">l</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">l</font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font>;<br/>





<b>set </b><font color="Maroon">F1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">ll</font> =  <a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">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#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span> ;<br/>
<a NAME="E1:59_1"/><i><font color="Green">E43</font></i>: 
(  <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  &amp;  <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T77">QC_LANG3:77</a></i>;<br/>
<a NAME="E2:59_1"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:59_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  implies <font color="Maroon">y</font> <a href="hidden.html#R2">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#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:59_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:59_1_1_1"/><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> )
 <b>and </b><br/><a NAME="E5:59_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a> 
 ;<br/>
<a NAME="E6:59_1_1_1"/>
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang3.html#K3">a.</a> 0
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:59"><i><font color="Green">E40</font></i></a>, <a class="ref" href="qc_lang3.html#T40">QC_LANG3:40</a></i>;<br/>
<a NAME="E7:59_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:59_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">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#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="cqc_lang.html#E9"><i><font color="Green">Lemma37</font></i></a>, , , <a class="txt" href="cqc_lang.html#E1:59"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:59_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">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#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:59_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E5:59_1_1"/><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E6:59_1_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_1_2"/>
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K3">a.</a> 0
 ;<br/><a NAME="E2:59_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="cqc_lang.html#E9"><i><font color="Green">Lemma37</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:59_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:59"><i><font color="Green">E40</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E8:59_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="cqc_lang.html#E9"><i><font color="Green">Lemma37</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E9:59_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="cqc_lang.html#E9"><i><font color="Green">Lemma37</font></i></a>, , , <a class="txt" href="cqc_lang.html#E1:59"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:59_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5" target="_self">Subst</a> <font color="Maroon">l</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E5:59_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:59"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9" target="_self">VERUM</a> ]
 
<b>by </b><i/>;<br/>
<a NAME="E3:59"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12">'not'</a> <font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:59_2"/><i><font color="Green">E70</font></i>: 
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">p</font>
 ;<br/>

<b>thus </b> <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="qc_lang3.html#T78">QC_LANG3:78</a></i>
<br/>.= 
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T78">QC_LANG3:78</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:59"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Olive">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:59_3"/><i><font color="Green">E72</font></i>: 
(  <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">p</font> &amp;  <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">q</font> )
 ;<br/>

<b>thus </b> <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i>, <a class="ref" href="qc_lang3.html#T80">QC_LANG3:80</a></i>
<br/>.= 
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T80">QC_LANG3:80</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:59"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14">All</a> <font color="Olive">y</font>,<font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:59_4"/><i><font color="Green">E126</font></i>: 
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><b>hence </b> <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">p</font>
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:59"><i><font color="Green">E40</font></i></a>, <a class="ref" href="qc_lang3.html#T81">QC_LANG3:81</a></i>
<br/>.= 
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T81">QC_LANG3:81</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59_4"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14">All</a> <font color="Maroon">y</font>,<font color="Maroon">p</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:59"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S1">QC_LANG1:sch 1</a>(, , <a class="txt" href="cqc_lang.html#E9"><i><font color="Green">Lemma37</font></i></a>, , );<br/></i>
<b>hence </b><a NAME="E7:59"/>
 <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">p</font>
 ;<br/>


</div>
