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

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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   <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</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="Olive" title="b1">p</font></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K13" title="CQC_LANG:func.13">.</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></span><br/>



<span class="kw">defpred </span><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> ] <span class="kw">means </span> <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#K13" title="CQC_LANG:func.13">.</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="E1:53"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
 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="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c3">p</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
 <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#K13" title="CQC_LANG:func.13">.</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c3">p</font>]</span><br/>

<span class="kw">thus </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"><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#K13" title="CQC_LANG:func.13">.</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#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T19" target="_self" title="CQC_LANG:th.19">Th32</a></span>
<br/>.= 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c3">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="qc_lang3.html#T65" title="QC_LANG3:th.65">QC_LANG3:65</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T65" title="QC_LANG3:th.65">QC_LANG3:65</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:53"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
 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="53_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">q</font>] implies <font color="Maroon">S<sub>1</sub></font>[<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><br/>



<span class="kw">assume </span><a NAME="E1:53_2"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
(  <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#K13" title="CQC_LANG:func.13">.</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#K13" title="CQC_LANG:func.13">.</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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<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><br/>

<span class="kw">thus </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="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#K13" title="CQC_LANG:func.13">.</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#K13" title="CQC_LANG:func.13">.</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#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T21" target="_self" title="CQC_LANG:th.21">Th34</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="qc_lang3.html#T67" title="QC_LANG3:th.67">QC_LANG3:67</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T67" title="QC_LANG3:th.67">QC_LANG3:67</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:53"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 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#NM7" title="QC_LANG1:NM.7">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="53_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c3">k</font><br/>  for <font color="Olive" title="b2">l</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c3">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b2">l</font>]</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c3">k</font> holds  <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="Olive" title="b1">l</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c5">l</font> be   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c3">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>]</span><br/>





<span class="kw">set </span><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/>
<span class="kw">set </span><font color="Maroon" title="c7">ll</font> =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span>);<br/>
<span class="kw">set </span><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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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="E1:53_3"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
 <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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#D1" target="_self" title="CQC_LANG:def.1">Def3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>  ) &amp; ( <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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>  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"><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>  ) )</span><br/><span class="kw">thus </span><a NAME="E1:53_3_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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>  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"><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>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:53_3_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:53_3_1_1"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:53_3_1_1"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
( 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> )
 <span class="kw">and </span><br/><a NAME="E5:53_3_1_1"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<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="E6:53_3_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_3_1_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="qc_lang3.html#T32" title="QC_LANG3:th.32">QC_LANG3:32</a></span>;<br/>
<a NAME="E7:53_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_3_1_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="cqc_lang.html#T3" target="_self" title="CQC_LANG:th.3">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:53_3_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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_3"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E3:53_3_1_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E4:53_3_1_1"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E5:53_3_1_1"><span class="lab"><font color="Green" title="E37">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:53_3_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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:53_3_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>
 <span class="kw">and </span><br/><a NAME="E5:53_3_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
( 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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span>)</span>)</span> )
 <span class="kw">and </span><br/><a NAME="E6:53_3_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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="53_3_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:53_3_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:53_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_3"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E5:53_3_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="cqc_lang.html#T3" target="_self" title="CQC_LANG:th.3">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E3:53_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:53_3_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="cqc_lang.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E8:53_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_3"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E5:53_3_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="cqc_lang.html#T3" target="_self" title="CQC_LANG:th.3">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E9:53_3_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_3"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E4:53_3_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E5:53_3_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E6:53_3_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:53_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<a NAME="E4:53_3"/>
(  <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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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#K1" title="CQC_LANG:func.1">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#K2" title="CQC_LANG:func.2">.--&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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T64" title="QC_LANG3:th.64">QC_LANG3:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:53_3"/>
<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>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53_3"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="cqc_lang.html#T17" target="_self" title="CQC_LANG:th.17">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:53"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
 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="53_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">p</font>)]</span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">p</font>] implies <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>)] )</span><br/>



<span class="kw">assume </span><a NAME="E1:53_4"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
 <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#K13" title="CQC_LANG:func.13">.</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:53_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K13" title="CQC_LANG:func.13">.</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> <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></span><br/><span class="kw">hence </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"><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#K13" title="CQC_LANG:func.13">.</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#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T25" target="_self" title="CQC_LANG:th.25">Th38</a></span>
<br/>.= 
 <a href="qc_lang3.html#K6" title="QC_LANG3:func.6">Fixed</a> <font color="Maroon" title="c4">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_4"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="qc_lang3.html#T68" title="QC_LANG3:th.68">QC_LANG3:68</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T68" title="QC_LANG3:th.68">QC_LANG3:68</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:53_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>)]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T24" target="_self" title="CQC_LANG:th.24">Th37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:53"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T15" target="_self" title="CQC_LANG:th.15">Th28</a></span>;<br/>
<a NAME="E6:53"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang1.html#S1" title="QC_LANG1:sch.1">QC_LANG1:sch 1</a>(<a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E34">A14</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E7:53"/>
 <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#K13" title="CQC_LANG:func.13">.</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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