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

<a NAME="E1:9_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E2:9_1_1"/><b>then </b><i><font color="Green">E3</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:9_1_1"/><i><font color="Green">E4</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> holds <br/>( ( <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> implies <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) &amp; ( not <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> implies <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:9_1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:9_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></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="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="substut1.html#E1:9_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1_1_1"/>
( ( <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> implies <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ) &amp; ( not <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> implies <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="ref" href="substut1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:9_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:9_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:9_1_1_2"/><i><font color="Green">E6</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="substut1.html#E3:9_1_1_2"><i><font color="Green">E5</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> or  not <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:9_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_2_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 <b>by </b><i><a class="txt" href="substut1.html#E2:9_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="substut1.html#E3:9_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="substut1.html#E3:9_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="substut1.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:9_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:9_1_1_2_1_1_2"/>
not <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:9_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E7</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="substut1.html#E2:9_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="substut1.html#E3:9_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="substut1.html#E3:9_1_1_2"><i><font color="Green">E5</font></i></a></i>;<br/><a NAME="E3:9_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="substut1.html#E2:9_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="substut1.html#E3:9_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:9_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E8</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E5:9_1_1_2_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 ;<br/><a NAME="E6:9_1_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 ;<br/><b>hence </b><a NAME="E7:9_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p2">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <b>by </b><i><a class="txt" href="substut1.html#E2:9_1_1_2_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="substut1.html#E4:9_1_1_2_1_1_2"><i><font color="Green">E8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:9_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <b>by </b><i><a class="txt" href="substut1.html#E4:9_1_1_2"><i><font color="Green">E6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:9_1_1"/>
 <a href="substut1.html#K3" target="_self">CQC_Subst</a> <span class="p1">(<span class="default"><a href="substut1.html#K4" target="_self">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="qc_lang1.html#K2">bound_QC-variables</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>


</div>
