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

<b>let </b><font color="Maroon">s1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> , <font color="Maroon">s2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:20_1_2"/><i><font color="Green">E49</font></i>: 
 ex <font color="Olive">pp</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="finseq_1.html#K13">*</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">pp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">pp</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<font color="Olive">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span> <a href="scmfsa_2.html#K17">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Olive">pp</font> )
 <b>and </b><br/><a NAME="E2:20_1_2"/><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">pp</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="finseq_1.html#K13">*</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">pp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Olive">pp</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<font color="Olive">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span> <a href="scmfsa_2.html#K17">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Olive">pp</font> )
 ;<br/>

<b>consider </b><font color="Maroon">pp1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E4:20_1_2"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:20_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">pp1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<font color="Olive">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span> <a href="scmfsa_2.html#K17">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>and </b><br/><a NAME="E6:20_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">pp1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">pp2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="ami_1.html#U4">Instructions</a> of <a href="scmfsa_2.html#K1">SCM+FSA</a>  <a href="finseq_1.html#K13">*</a> <b> such that </b><br/><a NAME="E8:20_1_2"/><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E9:20_1_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> &amp; <font color="Maroon">pp2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<font color="Olive">i</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span> <a href="scmfsa_2.html#K17">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>and </b><br/><a NAME="E10:20_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">pp2</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">i</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E12:20_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">pp1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:20_1_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pp2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:20_1_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E2:20_1_2_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:20_1_2_1_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">pp1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<font color="Maroon">i1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span> <a href="scmfsa_2.html#K17">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">i2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:20_1_2_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">pp2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" target="_self">aSeq</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span>,<font color="Maroon">i2</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 1</span>)</span> <a href="scmfsa_2.html#K17">:=</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K4">intloc</a> 2</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="scmfsa_7.html#T1" target="_self">Th1</a>, <a class="txt" href="scmfsa_7.html#E2"><i><font color="Green">Lemma26</font></i></a></i>;<br/><b>thus </b><a NAME="E7:20_1_2_1_1"/>
<font color="Maroon">pp1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:20_1_2"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div>
