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

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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="funct_1.html#V2">one-to-one</a> holds <br/> <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">q</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>assume </b><a NAME="E1:69_1"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>assume </b><a NAME="E2:69_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a> )
 ;<br/><a NAME="E3:69_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_4.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:69_1"/><b>then </b>
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/><b>hence </b><a NAME="E5:69_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:69"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:69_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="funct_1.html#V2">one-to-one</a> holds <br/> <a href="relat_1.html#K2">rng</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">q</font>
 ;<br/><b>let </b><font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:69_2"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E3:69_2"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E4:69_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">q</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/><a NAME="E5:69_2"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:69_2"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_2_1_1"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> or  not <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_2_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_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="E1:69_2_1_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_2_1_1_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_2_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_2_1_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">r</font> = <font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E2:69_2_1_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="finseq_4.html#E1:69_2_1_1_1_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_3.html#T94">FINSEQ_3:94</a></i>;<br/><i><font color="Green">E73</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T8" target="_self">Th8</a>, <a class="txt" href="finseq_4.html#E1:69_2_1_1_1_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_3.html#T97">FINSEQ_3:97</a></i>
<br/>.= 
<font color="Maroon">k</font>

;<br/><a NAME="E4:69_2_1_1_1_1_1_2"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T72">FINSEQ_3:72</a></i>;<br/><a NAME="E5:69_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T9" target="_self">Th9</a>, <a class="txt" href="finseq_4.html#E2:69_2_1_1_1_1_1_2"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/><a NAME="E6:69_2_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T12">FINSEQ_1:12</a></i>;<br/><a NAME="E7:69_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, </i>;<br/><a NAME="E8:69_2_1_1_1_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:69_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_4.html#E2:69_2_1_1_1_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_4.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:69_2_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T12">FINSEQ_1:12</a></i>;<br/><b>hence </b><a NAME="E11:69_2_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:69_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_2_1_1_2"/><i><font color="Green">E77</font></i>: 
not <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">q</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19">FINSEQ_1:19</a></i>;<br/><a NAME="E3:69_2_1_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T8" target="_self">Th8</a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/><a NAME="E4:69_2_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/><a NAME="E5:69_2_1_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:69_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:69_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T9" target="_self">Th9</a>, , <a class="ref" href="finseq_4.html#T11" target="_self">Th11</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:69_2_1_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:69_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T12">FINSEQ_1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:69_2_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:69_2_1_1_2"/>
<font color="Maroon">r</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/><a NAME="E8:69_2_1_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_4.html#D3" target="_self">Def3</a>, </i>;<br/><a NAME="E9:69_2_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E10:69_2_1_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:69_2_1_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E13:69_2_1_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E15:69_2_1_1_2"/><i><font color="Green">E108</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/><a NAME="E16:69_2_1_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E17:69_2_1_1_2"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_4.html#T17" target="_self">Th17</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E18:69_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font>
 <b>by </b><i>, , , , , <a class="ref" href="finseq_4.html#D4" target="_self">Def4</a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E7:69_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:69"/><b>then </b><i><font color="Green">E110</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:69"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="finseq_4.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_4.html#T7" target="_self">Th7</a>);<br/></i>
<a NAME="E6:69"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
;<br/>
<b>hence </b><a NAME="E7:69"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a> implies  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T8" target="_self">Th8</a></i>;<br/>


</div>
