<?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>let </b><font color="Maroon">p</font> be  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/><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>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> <a href="funct_7.html#K5" target="_self">apply</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:68"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></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">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E2:68"/><b>then </b>
 <a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:68"/><i><font color="Green">E30</font></i>: 
 <a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</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">y</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/>
<a NAME="E5:68"/>
 <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">y</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="E6:68"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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#E1:68"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_7.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E7:68"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></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">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
 <a href="funct_7.html#K5" target="_self">apply</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>,<font color="Maroon">x</font> = 
 <a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="funct_7.html#T3" target="_self">Th3</a>, <a class="txt" href="funct_7.html#E4:68"><i><font color="Green">E30</font></i></a>, <a class="ref" href="rewrite1.html#T2">REWRITE1:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E9:68"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E10:68"/><i><font color="Green">E44</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="68_2"><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="E1:68_2"/><i><font color="Green">E45</font></i>: 
 <a href="funct_7.html#K5" target="_self">apply</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>

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

<a NAME="E2:68_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</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/>;<br/>
<a NAME="E3:68_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <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">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E4:68_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</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 class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <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>by </b><i/>;<br/>
<a NAME="E5:68_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="funct_7.html#K5" target="_self">apply</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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <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">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E6:68_2"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font></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> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</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>
 
<b>by </b><i/>;<br/>
<b>thus </b> <a href="funct_7.html#K5" target="_self">apply</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</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>,<font color="Maroon">x</font> = 
 <a href="funct_7.html#K5" target="_self">apply</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span> <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>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">q</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="funct_7.html#T3" target="_self">Th3</a>, <a class="txt" href="funct_7.html#E9:68"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_7.html#T6" target="_self">Th6</a>, <a class="ref" href="rewrite1.html#T2">REWRITE1:2</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">q</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</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 class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</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 class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T3" target="_self">Th3</a>, , <a class="ref" href="rewrite1.html#T2">REWRITE1:2</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E11:68"/>
for <font color="Olive">q</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">q</font>]
 
<b>from </b><i><a class="ref" href="funct_7.html#S5" target="_self">FUNCT_7:sch 5</a>(<a class="txt" href="funct_7.html#E7:68"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_7.html#T4" target="_self">Th4</a>);<br/></i>
<b>hence </b><a NAME="E12:68"/>
 <a href="funct_7.html#K5" target="_self">apply</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K5" target="_self">apply</a> <font color="Maroon">p</font>,<font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>


</div>
