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

<b>set </b><font color="Maroon">A</font> =  <a href="dtconstr.html#K4">TS</a> <span class="p1">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="freealg.html#K13" target="_self">FreeOpZAO</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font>,<font color="Maroon">D</font>;<br/>
<a NAME="E1:36_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</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> <font color="Maroon">f</font></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36_1_1_1"/>
<font color="Maroon">n</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> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">O</font> =  <a href="freealg.html#K13" target="_self">FreeOpZAO</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<font color="Maroon">O</font>
;<br/>

<b>thus </b><a NAME="E3:36_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">O</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p3">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> </span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <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> <font color="Maroon">f</font></span>)</span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</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> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>] ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>();<br/></i>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="unialg_1.html#NM1">PFuncFinSequence</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" target="_self">DTConUA</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E5:36_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E6:36_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>hence </b><a NAME="E7:36_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="freealg.html#K13" target="_self">FreeOpZAO</a> <font color="Maroon">n</font>,<font color="Maroon">f</font>,<font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div>
