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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="finseq_1.html#NM4">FinSubsequence</a>;<br/>





<b>assume </b><a NAME="E1:111"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>1</sub></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_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="finseq_1.html#NM4">FinSubsequence</a><b> such that </b><br/><a NAME="E3:111"/><i><font color="Green">E75</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &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="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</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">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="pnproc_1.html#T56" target="_self">Th56</a></i>;<br/>
<a NAME="E4:111"/><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E3:111"><i><font color="Green">E75</font></i></a>, <a class="txt" href="pnproc_1.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/>
<a NAME="E5:111"/><i><font color="Green">E77</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T71">FINSEQ_1:71</a></i>;<br/>
<i><font color="Green">E78</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:111"><i><font color="Green">E77</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/>
<a NAME="E7:111"/><i><font color="Green">E79</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#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:111_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>5</sub></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_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:111_2"/><i><font color="Green">E81</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E6:111"><i><font color="Green">E78</font></i></a>, <a class="txt" href="pnproc_1.html#E1:111_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:111_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T71">FINSEQ_1:71</a></i>;<br/>
<a NAME="E4:111_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="pnproc_1.html#E2:111_2"><i><font color="Green">E81</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="pnproc_1.html#E3:111"><i><font color="Green">E75</font></i></a>, <a class="txt" href="pnproc_1.html#E6:111"><i><font color="Green">E78</font></i></a>, <a class="txt" href="pnproc_1.html#E1:111_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="pnproc_1.html#E3:111"><i><font color="Green">E75</font></i></a>, <a class="txt" href="pnproc_1.html#E3:111_2"><i><font color="Green">E82</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E7:111_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:111"/><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E3:111"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#T71">FINSEQ_1:71</a></i>;<br/>
<a NAME="E9:111"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E10:111"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E8:111"><i><font color="Green">E80</font></i></a>, <a class="txt" href="pnproc_1.html#E14"><i><font color="Green">Lemma14</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:111"/><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/>
<i><font color="Green">E82</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pnproc_1.html#E3:111"><i><font color="Green">E75</font></i></a>, <a class="txt" href="pnproc_1.html#E5:111"><i><font color="Green">E77</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
;<br/>
<a NAME="E15:111"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:111_4"/><i><font color="Green">E83</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<i><font color="Green">E84</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
 <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>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> ) </span>}</span> 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>
;<br/>
<a NAME="E4:111_4"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E1:111_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E5:111_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E1:111_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:111_4"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>11</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>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:111_4"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 0 <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>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:111_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E9:111_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E6:111_4"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:111_4"/><b>then </b><i><font color="Green">E86</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E6:111"><i><font color="Green">E78</font></i></a>, <a class="txt" href="pnproc_1.html#E14:111"><i><font color="Green">E82</font></i></a>, <a class="txt" href="pnproc_1.html#E1:111_4"><i><font color="Green">E83</font></i></a>, <a class="txt" href="pnproc_1.html#E3:111_4"><i><font color="Green">E84</font></i></a></i>;<br/>
<a NAME="E11:111_4"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E1:111_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:111_4"/><b>then </b><i><font color="Green">E87</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"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E6:111"><i><font color="Green">E78</font></i></a>, <a class="txt" href="pnproc_1.html#E14:111"><i><font color="Green">E82</font></i></a>, <a class="txt" href="pnproc_1.html#E1:111_4"><i><font color="Green">E83</font></i></a>, <a class="txt" href="pnproc_1.html#E3:111_4"><i><font color="Green">E84</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:111_4"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</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="nat_1.html#K1">+</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E7:111"><i><font color="Green">E79</font></i></a>, <a class="txt" href="pnproc_1.html#E10:111_4"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:111_4"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E7:111"><i><font color="Green">E79</font></i></a>, <a class="txt" href="pnproc_1.html#E12:111_4"><i><font color="Green">E87</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E18:111_4"/><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/>
<a NAME="E19:111_4"/>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</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>5</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:111_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E14:111"><i><font color="Green">E82</font></i></a>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/>
<a NAME="E21:111_4"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E1:111_4"><i><font color="Green">E83</font></i></a>, <a class="txt" href="pnproc_1.html#E14:111_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="pnproc_1.html#E16:111_4"><i><font color="Green">E89</font></i></a>, <a class="txt" href="pnproc_1.html#E18:111_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E22:111_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E1:111_4"><i><font color="Green">E83</font></i></a>, <a class="txt" href="pnproc_1.html#E14:111_4"><i><font color="Green">E88</font></i></a>, <a class="txt" href="pnproc_1.html#E16:111_4"><i><font color="Green">E89</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:111"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E4:111"><i><font color="Green">E76</font></i></a>, <a class="txt" href="pnproc_1.html#E13:111"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E17:111"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pnproc_1.html#K14" target="_self">Shift</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E1:111"><i><font color="Green">E74</font></i></a>, <a class="txt" href="pnproc_1.html#E7:111"><i><font color="Green">E79</font></i></a></i>;<br/>


</div>
