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

<b>let </b><font color="Maroon">fs</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:55"/><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font>
 ;<br/>

<b>set </b><font color="Maroon">fs1</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span>;<br/>
<b>set </b><font color="Maroon">fs2</font> = <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>;<br/>
<a NAME="E2:55"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green">E48</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:55"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:55"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">fs</font>
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:55"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:55"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:55"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
;<br/>
<i><font color="Green">E105</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<i><font color="Green">E106</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a>, <a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E9:55"/>
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> holds <br/><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:55_3"/><i><font color="Green">E107</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_3_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="wsierp_1.html#E4:55"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1_1"><b>suppose </b></a><a NAME="E1:55_3_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:55_3_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="wsierp_1.html#E5:55"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E5:55"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1_2"><b>suppose </b></a><a NAME="E1:55_3_1_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:55_3_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E4:55_3_1_1_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a>, <a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, <a class="txt" href="wsierp_1.html#E4:55"><i><font color="Green">E49</font></i></a>, <a class="txt" href="wsierp_1.html#E5:55"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>
<br/>.= 
<font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T5" target="_self">Th5</a>, , </i>
<br/>.= 
<font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>

;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_3"/>
<font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:55"/>
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>

<a NAME="E11:55"/>
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> holds <br/><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:55_4"/><i><font color="Green">E113</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_4_1_1"/>
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font> )
 </span><b>by </b><i><a class="txt" href="wsierp_1.html#E4:55"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_4_1_1_1"><b>suppose </b></a><a NAME="E1:55_4_1_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K5">-</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E3:55_4_1_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="txt" href="wsierp_1.html#E5:55"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a>, <a class="txt" href="wsierp_1.html#E5:55"><i><font color="Green">E50</font></i></a>, <a class="txt" href="wsierp_1.html#E3:55"><i><font color="Green">E48</font></i></a>, <a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E5:55"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_4_1_1_2"><b>suppose </b></a><a NAME="E1:55_4_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font>
 ;<br/><div class="add"><a NAME="E2:55_4_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:55_4_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:55_4_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E4:55"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, <a class="txt" href="wsierp_1.html#E5:55"><i><font color="Green">E50</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_4"/>
<font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:55"/>
<span class="p1">(<span class="default"><a href="finseq_3.html#K2">Del</a> <font color="Maroon">fs</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">fs</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">fs</font></span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">fs</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E3:55"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
