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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is   <a href="funct_7.html#NM1">FuncSequence</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K6" target="_self">firstdom</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> );<br/>
<a NAME="E1:85"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E2:85"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">p</font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/>for <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Olive">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:85_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">q</font> is   <a href="funct_7.html#NM1">FuncSequence</a> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <font color="Maroon">q</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K6" target="_self">firstdom</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E2:85_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> is   <a href="funct_7.html#NM1">FuncSequence</a> &amp; <font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:85_1"/><i><font color="Green">E44</font></i>: 
 <a href="funct_7.html#K4" target="_self">compose</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <font color="Maroon">q</font>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1_1_1"><b>suppose </b></a><a NAME="E1:85_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:85_1_1_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E3:85_1_1_1"/><b>then </b>
(  <a href="funct_7.html#K4" target="_self">compose</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="funct_7.html#K6" target="_self">firstdom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="funct_7.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E4:85_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K6" target="_self">firstdom</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T37">FUNCT_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_1_1_2"><b>suppose </b></a><a NAME="E1:85_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:85_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63">FINSEQ_1:63</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:85_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="funct_7.html#E3:85_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="rewrite1.html#T5">REWRITE1:5</a></i>;<br/><a NAME="E6:85_1_1_2"/>
( <font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">y</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="funct_7.html#E3:85_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/><a NAME="E7:85_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="funct_7.html#K6" target="_self">firstdom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K1">proj1</a> <font color="Maroon">y</font> &amp;  <a href="funct_7.html#K6" target="_self">firstdom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K1">proj1</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="funct_7.html#E3:85_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="funct_7.html#E1:85_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:85_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E10:85_1_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E11:85_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E12:85_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E13:85_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i><a class="txt" href="funct_7.html#E3:85_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E14:85_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="funct_7.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:85_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="funct_7.html#E7:85_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E16:85_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E17:85_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="funct_7.html#E1:85_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E18:85_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="funct_7.html#E1:85_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:85_1_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E21:85_1_1_2"/><i><font color="Green">E95</font></i>: 
 ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="funct_7.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E22:85_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="funct_7.html#K7" target="_self">lastrng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="txt" href="funct_7.html#E1:85_1_1_2"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="funct_7.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E23:85_1_1_2"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <font color="Maroon">q</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K6" target="_self">firstdom</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <font color="Maroon">q</font>,<font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_7.html#K7" target="_self">lastrng</a> <font color="Maroon">q</font> )
 <b>by </b><i>, , <a class="txt" href="funct_7.html#E3:85_1"><i><font color="Green">E44</font></i></a>, , </i>;<br/><a NAME="E24:85_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <font color="Maroon">q</font>,<font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="funct_7.html#E12:85_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E25:85_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K6" target="_self">firstdom</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <span class="p3"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">f</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T11" target="_self">Th11</a>, <a class="txt" href="funct_7.html#E3:85_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="funct_7.html#E5:85_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="funct_7.html#E10:85_1_1_2"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:85"/>
for <font color="Olive">p</font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="funct_7.html#S5" target="_self">FUNCT_7:sch 5</a>(, );<br/></i>
<b>hence </b><a NAME="E4:85"/>
for <font color="Olive">p</font> being  <a href="funct_7.html#NM1">FuncSequence</a>  st <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4" target="_self">compose</a> <font color="Olive">p</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K6" target="_self">firstdom</a> <font color="Olive">p</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 ;<br/>


</div>
