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

<b>let </b><font color="Maroon">fp</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> 1;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">fr</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">fr</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">fr1</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">fr1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:74"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5" target="_self">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E2:74"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="wsierp_1.html#K5" target="_self">&lt;*&gt;</a> <a href="numbers.html#K5">NAT</a> ]
 
;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">fp</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">fp</font>]
 ;<br/><b>set </b><font color="Maroon">fp1</font> = <font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><b>set </b><font color="Maroon">k</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_1_1"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>] )
 ;<br/><a NAME="E2:74_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&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">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> 2 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 )
 </span><b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_1"/><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">fr</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> ;<br/><b>assume </b><a NAME="E1:74_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fr</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E2:74_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:74_1_1_1_1_1_1_1"/>
( 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">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E2:74_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:74_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 2</span>)</span> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T16">INT_2:16</a></i>;<br/><a NAME="E4:74_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 2</span>)</span> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 2</span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">m</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:74_1_1_1_1_1_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:74_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font>, <font color="Maroon">y</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>take </b><font color="Maroon">fr1</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/><b>thus </b><a NAME="E8:74_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fr1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E9:74_1_1_1_1_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">b</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">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E10:74_1_1_1_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">b</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"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1_1_1_1_1_2_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="seq_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_1_1_2_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_1_1_2_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">fr1</font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font> &amp; <font color="Maroon">fr1</font> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E3:74_1_1_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E19"><i><font color="Green">Lemma58</font></i></a></i>;<br/><b>hence </b><a NAME="E4:74_1_1_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="seq_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="wsierp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T34">XCMPLX_1:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:74_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="seq_1.html#K2">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:74_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_2"/><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E3:74_1_1_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:74_1_1_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">fp</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:74_1_1_1_1_2_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E2:74_1_1_1_1_2_1"/><i><font color="Green">E116</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E3:74_1_1_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:74_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="wsierp_1.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">fr2</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> ;<br/><b>assume </b><a NAME="E1:74_1_1_1_1_2_2"/><i><font color="Green">E123</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fr2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E2:74_1_1_1_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fr2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma57</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">fr</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> , <font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E4:74_1_1_1_1_2_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">fr2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fr</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">m</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E6:74_1_1_1_1_2_2"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</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">fr</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T10" target="_self">Th10</a>, <a class="ref" href="wsierp_1.html#T13" target="_self">Th13</a>, <a class="ref" href="wsierp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E7:74_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fr</font>
 ;<br/><b>then consider </b><font color="Maroon">fr1</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a> <b> such that </b><br/><a NAME="E9:74_1_1_1_1_2_2"/><i><font color="Green">E127</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fr1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font> &amp; ( for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E15"><i><font color="Green">Lemma54</font></i></a>, <a class="txt" href="wsierp_1.html#E19"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="wsierp_1.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:74_1_1_1_1_2_2"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">a</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_1_1_1_2_2_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font>
 ;<br/>

<a NAME="E2:74_1_1_1_1_2_2_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E3:74_1_1_1_1_2_2_1"/><i><font color="Green">E145</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E4:74_1_1_1_1_2_2_1"/><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E5:74_1_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:74_1_1_1_1_2_2_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:74_1_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</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">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>hence </b><a NAME="E8:74_1_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T9" target="_self">Th9</a>, <a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a>, <a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="wsierp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:74_1_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a></i>;<br/><a NAME="E12:74_1_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><a NAME="E13:74_1_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">d</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T16">INT_2:16</a></i>;<br/><b>then consider </b><font color="Maroon">m1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">n1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E15:74_1_1_1_1_2_2"/><i><font color="Green">E153</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:74_1_1"><i><font color="Green">E50</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K10" target="_self">Del</a> <font color="Maroon">fp</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">s2</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E17:74_1_1_1_1_2_2"/><i><font color="Green">E156</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font> &amp; ( for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</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">fp</font></span>)</span> holds <br/><font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i><a NAME="E18:74_1_1_1_1_2_2"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s2</font> holds <br/><font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K2">INT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_1_1_1_2_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s2</font>
 ;<br/>

<a NAME="E2:74_1_1_1_1_2_2_2"/><b>then </b>
<font color="Maroon">a</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">s2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:74_1_1_1_1_2_2_2"/><b>then </b>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K10" target="_self">Del</a> <font color="Maroon">fp</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:74_1_1_1_1_2_2_2"/>
<font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K2">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">s2</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T14">FINSEQ_2:14</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="mesfunc1.html#K2">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/><b>take </b><font color="Maroon">fr3</font> = <font color="Maroon">s2</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><b>thus </b><a NAME="E21:74_1_1_1_1_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fr3</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T10" target="_self">Th10</a>, <a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E22:74_1_1_1_1_2_2"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">b</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">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><b>thus </b><a NAME="E23:74_1_1_1_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_2_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:74_1_1_1_1_2_2_3"/><i><font color="Green">E159</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T10" target="_self">Th10</a>, <a class="ref" href="wsierp_1.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E2:74_1_1_1_1_2_2_3"/><i><font color="Green">E160</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">c</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">fp</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">c</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_2_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:74_1_1_1_1_2_2_3_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">c</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">fp</font></span>)</span>
 ;<br/>

<a NAME="E2:74_1_1_1_1_2_2_3_1"/><b>then </b><i><font color="Green">E162</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s2</font> )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma67</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:74_1_1_1_1_2_2_3_1"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c</font> = 
<font color="Maroon">s2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K10" target="_self">Del</a> <font color="Maroon">fp</font>,<font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K10" target="_self">Del</a> <font color="Maroon">fp</font>,<font color="Maroon">c</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E35"><i><font color="Green">Lemma73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="wsierp_1.html#E3:74"><i><font color="Green">E41</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1_1_2_2_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_2_2_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1_1_1_2_2_3_2_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font> or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_2_2_3_2_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_2_2_3_2_1_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font>
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_2_2_3_2_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:74_1_1_1_1_2_2_3_2_1_1"/><b>then </b><i><font color="Green">E165</font></i>: 
( 1 <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">fp</font></span>)</span> &amp; <font color="Maroon">b</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">fp</font></span>)</span> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T20" target="_self">Th20</a>, <a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a></i>;<br/><a NAME="E4:74_1_1_1_1_2_2_3_2_1_1"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fr</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fr</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:74_1_1_1_1_2_2_3_2_1_1"/><b>then </b><i><font color="Green">E166</font></i>: 
( <font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1 &amp; <font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:74_1_1_1_1_2_2_3_2_1_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a>, <a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E35"><i><font color="Green">Lemma73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T16" target="_self">Th16</a>, <a class="txt" href="wsierp_1.html#E36"><i><font color="Green">Lemma74</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a>, <a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="wsierp_1.html#E35"><i><font color="Green">Lemma73</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1_2_2_3_2_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_1_1_2_2_3_2_1_2"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:74_1_1_1_1_2_2_3_2_1_2"/><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="wsierp_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E3:74_1_1_1_1_2_2_3_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T11" target="_self">Th11</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:74_1_1_1_1_2_2_3_2_1_2"/><b>then </b>
1 <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">fp</font></span>)</span>
 ;<br/><b>then </b><i><font color="Green">E170</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8" target="_self">Product</a> <font color="Maroon">fp</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E34"><i><font color="Green">Lemma72</font></i></a></i>
;<br/><a NAME="E6:74_1_1_1_1_2_2_3_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="wsierp_1.html#T14" target="_self">Th14</a>, <a class="ref" href="wsierp_1.html#T15" target="_self">Th15</a>, <a class="txt" href="wsierp_1.html#E27"><i><font color="Green">Lemma67</font></i></a>, <a class="txt" href="wsierp_1.html#E32"><i><font color="Green">Lemma71</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><b>hence </b><a NAME="E7:74_1_1_1_1_2_2_3_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E35"><i><font color="Green">Lemma73</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="E4:74_1_1_1_1_2_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p4"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">fr3</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">fr2</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:74_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:74_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">fp</font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:74"/>
for <font color="Olive">fp</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">fp</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, <a class="txt" href="wsierp_1.html#E14"><i><font color="Green">Lemma53</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:74"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; ( for <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a> 1 ) implies for <font color="Olive">fr</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>   st  <a href="finseq_1.html#K3">len</a> <font color="Olive">fr</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font> holds <br/> ex <font color="Olive">fr1</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="mesfunc1.html#K2">INT</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">fr1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fp</font> &amp; ( for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fp</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">fp</font> <a href="wsierp_1.html#K3" target="_self">.</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">fr1</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">fr</font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> ) ) )
 ;<br/>


</div>
