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

<b>let </b><font color="Maroon">c<sub>1</sub></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> ] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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<sub>2</sub></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<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></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<sub>1</sub></font></span>)</span> <a href="nat_1.html#K8">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">b<sub>2</sub></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> ] means for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></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<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></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<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>3</sub></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">b<sub>2</sub></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">b<sub>1</sub></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> ] means (  <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="E1:75"/>
 <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="E2:75"/><b>then </b><i><font color="Green">E52</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">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:75_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1"/><i><font color="Green">E55</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>] )
 ;<br/><a NAME="E2:75_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></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="75_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_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="E1:75_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>3</sub></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="txt" href="wsierp_1.html#E1:75_1_1"><i><font color="Green">E55</font></i></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="75_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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="75_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a> ;<br/><b>assume </b><a NAME="E1:75_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E2:75_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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="75_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:75_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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>3</sub></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#E1:75_1_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E2:75_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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="txt" href="wsierp_1.html#E1:75_1_1"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:75_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>6</sub></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">c<sub>6</sub></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="E4:75_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>6</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma13</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:75_1_1_1_1_1_1"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>7</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T40" target="_self">Th40</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6" target="_self">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">c<sub>11</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/><b>thus </b><a NAME="E8:75_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E9:75_1_1_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E10:75_1_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9:75_1_1_1_1_1_1"><i><font color="Green">E59</font></i></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="75_1_1_1_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_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="E1:75_1_1_1_1_1_1_2_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="wsierp_1.html#E10:75_1_1_1_1_1_1"><i><font color="Green">E60</font></i></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="75_1_1_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_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">c<sub>2</sub></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">c<sub>3</sub></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<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>11</sub></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">c<sub>6</sub></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="75_1_1_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_1_1_2_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:75_1_1_1_1_1_1_2_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>8</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>7</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K4" target="_self">^</a> <span class="p5"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>7</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E6:75_1_1_1_1_1_1"><i><font color="Green">E58</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:75_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">c<sub>2</sub></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">c<sub>3</sub></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<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>11</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_1_1_2_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_1_1_2_1_2"><i><font color="Green">E62</font></i></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="E12:75_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">c<sub>2</sub></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">c<sub>3</sub></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<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>11</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:75_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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="75_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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="E2:75_1_1_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></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="E3:75_1_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></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="E4:75_1_1_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:75_1_1_1_1_2_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E2:75_1_1_1_1_2_1"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>3</sub></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="E3:75_1_1_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:75_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2_1"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a> ;<br/><b>assume </b><a NAME="E1:75_1_1_1_1_2_2"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><a NAME="E2:75_1_1_1_1_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a> , <font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a> <b> such that </b><br/><a NAME="E4:75_1_1_1_1_2_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></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">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E6:75_1_1_1_1_2_2"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="wsierp_1.html#E4:75_1_1_1_1_2_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><a NAME="E7:75_1_1_1_1_2_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a> <b> such that </b><br/><a NAME="E9:75_1_1_1_1_2_2"/><i><font color="Green">E65</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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="Maroon">c<sub>10</sub></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">c<sub>7</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="wsierp_1.html#E4:75_1_1_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:75_1_1_1_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:75_1_1_1_1_2_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:75_1_1_1_1_2_2_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E3:75_1_1_1_1_2_2_1"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>3</sub></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="E4:75_1_1_1_1_2_2_1"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E5:75_1_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:75_1_1_1_1_2_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:75_1_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>hence </b><a NAME="E8:75_1_1_1_1_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="wsierp_1.html#E3:75_1_1_1_1_2_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="wsierp_1.html#E4:75_1_1_1_1_2_2_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="wsierp_1.html#E6:75_1_1_1_1_2_2_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:75_1_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="wsierp_1.html#T43" target="_self">Th43</a></i>;<br/><a NAME="E12:75_1_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E18"><i><font color="Green">Lemma13</font></i></a></i>;<br/><a NAME="E13:75_1_1_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>3</sub></font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>10</sub></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">c<sub>6</sub></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">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E15:75_1_1_1_1_2_2"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>10</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T40" target="_self">Th40</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">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#K9" target="_self">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K11" target="_self">Del</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></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">c<sub>13</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E17:75_1_1_1_1_2_2"/><i><font color="Green">E67</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i><a NAME="E18:75_1_1_1_1_2_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="wsierp_1.html#K6" target="_self">INT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:75_1_1_1_1_2_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:75_1_1_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></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">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:75_1_1_1_1_2_2_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K11" target="_self">Del</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E17:75_1_1_1_1_2_2"><i><font color="Green">E67</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:75_1_1_1_1_2_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="wsierp_1.html#K6" target="_self">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">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">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">c<sub>15</sub></font> =  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="wsierp_1.html#K6" target="_self">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">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></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">c<sub>15</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><b>thus </b><a NAME="E21:75_1_1_1_1_2_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="wsierp_1.html#E17:75_1_1_1_1_2_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E22:75_1_1_1_1_2_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>17</sub></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">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/><b>thus </b><a NAME="E23:75_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">c<sub>2</sub></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">c<sub>3</sub></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<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>16</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_2_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:75_1_1_1_1_2_2_3"/><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></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">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="wsierp_1.html#E22:75_1_1_1_1_2_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E2:75_1_1_1_1_2_2_3"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></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#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_2_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:75_1_1_1_1_2_2_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>18</sub></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">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:75_1_1_1_1_2_2_3_1"/><b>then </b><i><font color="Green">E72</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E17:75_1_1_1_1_2_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:75_1_1_1_1_2_2_3_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<font color="Maroon">c<sub>16</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2_2_3_1"><i><font color="Green">E72</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#K9" target="_self">Product</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K11" target="_self">Del</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E17:75_1_1_1_1_2_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_3_1"><i><font color="Green">E71</font></i></a></i>
;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K11" target="_self">Del</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E3:75_1_1_1_1_2_2_3_1"><i><font color="Green">E73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2_2_3_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="wsierp_1.html#E52"><i><font color="Green">Lemma41</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_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="75_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="E1:75_1_1_1_1_2_2_3_2_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_3"><i><font color="Green">E69</font></i></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="75_1_1_1_1_2_2_3_2_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_2_2_3_2_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:75_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">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:75_1_1_1_1_2_2_3_2_1_1"/><b>then </b><i><font color="Green">E72</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">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>17</sub></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">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_3_2_1_1"><i><font color="Green">E71</font></i></a></i>;<br/><a NAME="E4:75_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">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E7:75_1_1_1_1_2_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:75_1_1_1_1_2_2_3_2_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:75_1_1_1_1_2_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E6:75_1_1_1_1_2_2_3_2_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:75_1_1_1_1_2_2_3_2_1_1"><i><font color="Green">E72</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">c<sub>2</sub></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">c<sub>3</sub></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<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></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#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2_2_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="wsierp_1.html#E3:75_1_1_1_1_2_2_3_2_1_1"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E5:75_1_1_1_1_2_2_3_2_1_1"><i><font color="Green">E73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></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">c<sub>10</sub></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">c<sub>7</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E9:75_1_1_1_1_2_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="wsierp_1.html#E6:75_1_1_1_1_2_2_3_2_1_1"><i><font color="Green">E74</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#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></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">c<sub>10</sub></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">c<sub>7</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>16</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2_2_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="wsierp_1.html#E3:75_1_1_1_1_2_2_3_2_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="wsierp_1.html#E5:75_1_1_1_1_2_2_3_2_1_1"><i><font color="Green">E73</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_2_2_3_2_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_1_1_2_2_3_2_1_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><i><font color="Green">E72</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>2</sub></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">c<sub>10</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>10</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>12</sub></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#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_3_2_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="wsierp_1.html#E15:75_1_1_1_1_2_2"><i><font color="Green">E66</font></i></a></i>
;<br/><a NAME="E3:75_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">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:75_1_1_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:75_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">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then </b><i><font color="Green">E73</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>16</sub></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">c<sub>6</sub></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#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></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">c<sub>10</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2_2_3"><i><font color="Green">E70</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K9" target="_self">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>11</sub></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">c<sub>2</sub></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">c<sub>10</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:75_1_1_1_1_2_2_3_2_1_2"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>

;<br/><a NAME="E6:75_1_1_1_1_2_2_3_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:75_1_1_1_1_2_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="wsierp_1.html#E6:75_1_1_1_1_2_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="wsierp_1.html#E17:75_1_1_1_1_2_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="wsierp_1.html#E1:75_1_1_1_1_2_2_3_2_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/><b>hence </b><a NAME="E7:75_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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>16</sub></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">c<sub>6</sub></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">c<sub>2</sub></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">c<sub>3</sub></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<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:75_1_1_1_1_2_2_3_2_1_2"><i><font color="Green">E73</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="E4:75_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">c<sub>2</sub></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">c<sub>3</sub></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<sub>17</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>17</sub></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">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>16</sub></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">c<sub>6</sub></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="E6:75_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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="E4:75_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:75_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:75"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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#E2:75"><i><font color="Green">E52</font></i></a>, <a class="txt" href="wsierp_1.html#E3:75"><i><font color="Green">E53</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:75"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="wsierp_1.html#K6" target="_self">INT</a>  st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<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">b<sub>2</sub></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">b<sub>1</sub></font> <a href="wsierp_1.html#K1" target="_self">.</a> 1</span>)</span> ) ) )
 ;<br/>


</div>
