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

<b>let </b><font color="Maroon">c<sub>1</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="E1:91"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</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="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>4</sub></font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>5</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  or <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> or <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E2:91"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</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="E3:91"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>4</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/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E4:91"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <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> : ( <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> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<b>assume </b><a NAME="E5:91"/><i><font color="Green">E64</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></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="E7:91"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_3.html#T59" target="_self">Th59</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <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> : ( <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>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> ;<br/>
<a NAME="E8:91"/><i><font color="Green">E66</font></i>: 
<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> : ( <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>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:91_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <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> : ( <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>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:91_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <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>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:91_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <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> : ( <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>6</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="finseq_3.html#E8:91"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_2_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_1"><b>suppose </b></a><a NAME="E1:91_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:91_2_1_1"/><b>then </b>
( not <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_3.html#E3:91"><i><font color="Green">E62</font></i></a>, <a class="txt" href="finseq_3.html#E5:91"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:91_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</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">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</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/><a NAME="E4:91_2_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T12">FINSEQ_1:12</a></i>;<br/><a NAME="E5:91_2_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:91_2_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E7:91_2_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_3.html#E1:91_2_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_3.html#T90" target="_self">Th90</a></i>;<br/><a NAME="E8:91_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:91_2_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:91_2_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:91_2_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:91_2_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:91_2_1_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:91_2_1_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<a NAME="E7:91_2_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="finseq_3.html#E6:91_2_1_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:91_2_1_1_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<b>hence </b><a NAME="E9:91_2_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E4:91"><i><font color="Green">E63</font></i></a>, <a class="txt" href="finseq_3.html#E3:91_2_1_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="finseq_3.html#E4:91_2_1_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="finseq_3.html#E5:91_2_1_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="finseq_3.html#E7:91_2_1_1_1_1"><i><font color="Green">E76</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>9</sub></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="E2:91_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:91_2_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E5:91_2_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>10</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>and </b><br/><a NAME="E6:91_2_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E7:91_2_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E4:91"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E8:91_2_1_1_1"/>
<font color="Maroon">c<sub>10</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="finseq_3.html#E5:91_2_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:91_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:91_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E6:91_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:91_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:91_2_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:91_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E7:91_2_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E14:91_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="finseq_3.html#E6:91_2_1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="finseq_3.html#E12:91_2_1_1_1"><i><font color="Green">E76</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:91_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:91"><i><font color="Green">E60</font></i></a>, <a class="txt" href="finseq_3.html#E5:91"><i><font color="Green">E64</font></i></a>, <a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="txt" href="finseq_3.html#E5:91_2_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="finseq_3.html#E6:91_2_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="finseq_3.html#E7:91_2_1_1"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_2"><b>suppose </b></a><a NAME="E1:91_2_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:91_2_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_3.html#T91" target="_self">Th91</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_2_1_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:91_2_1_2_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:91_2_1_2_1_1_1"/><i><font color="Green">E69</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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</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="E3:91_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:91_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:91_2_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:91_2_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>9</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>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E4:91_2_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_3.html#E1:91_2_1_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="finseq_3.html#E2:91_2_1_2_1_1_1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:91_2_1_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E6:91_2_1_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E4:91"><i><font color="Green">E63</font></i></a>, <a class="txt" href="finseq_3.html#E2:91_2_1_2_1_1_1_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>9</sub></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="E2:91_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E3:91_2_1_2_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <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> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E4:91"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:91_2_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:91_2_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_3.html#E1:91_2_1_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_3.html#T66" target="_self">Th66</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_3.html#E2:91_2_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="finseq_3.html#E2:91_2_1_2_1_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="finseq_3.html#E2:91_2_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><b>hence </b><a NAME="E6:91_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:91_2_1_2_1_1_1"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:91_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:91_2_1_2_1_1_2"/><b>then </b>
( not <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:91"><i><font color="Green">E61</font></i></a>, <a class="txt" href="finseq_3.html#E3:91"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:91_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</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">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</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/><a NAME="E4:91_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T12">FINSEQ_1:12</a></i>;<br/><a NAME="E5:91_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:91_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E7:91_2_1_2_1_1_2"/><i><font color="Green">E71</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E5:91"><i><font color="Green">E64</font></i></a>, <a class="txt" href="finseq_3.html#E5:91_2_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E8:91_2_1_2_1_1_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:91"><i><font color="Green">E60</font></i></a>, <a class="txt" href="finseq_3.html#E5:91"><i><font color="Green">E64</font></i></a>, <a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="txt" href="finseq_3.html#E5:91_2_1_2_1_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="finseq_3.html#E6:91_2_1_2_1_1_2"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E9:91_2_1_2_1_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:91_2_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></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="E1:91_2_1_2_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:91_2_1_2_1_1_2_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:91_2_1_2_1_1_2_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:91_2_1_2_1_1_2_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:91_2_1_2_1_1_2_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<a NAME="E7:91_2_1_2_1_1_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_2_1_1_2_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="finseq_3.html#E6:91_2_1_2_1_1_2_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E8:91_2_1_2_1_1_2_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<b>hence </b><a NAME="E9:91_2_1_2_1_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E4:91"><i><font color="Green">E63</font></i></a>, <a class="txt" href="finseq_3.html#E3:91_2_1_2_1_1_2_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="finseq_3.html#E4:91_2_1_2_1_1_2_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="finseq_3.html#E5:91_2_1_2_1_1_2_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="finseq_3.html#E7:91_2_1_2_1_1_2_1_1"><i><font color="Green">E78</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>9</sub></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="E2:91_2_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:91_2_1_2_1_1_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E5:91_2_1_2_1_1_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>10</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>and </b><br/><a NAME="E6:91_2_1_2_1_1_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E7:91_2_1_2_1_1_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E4:91"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E8:91_2_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>10</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="finseq_3.html#E5:91_2_1_2_1_1_2_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:91_2_1_2_1_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:91_2_1_2_1_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E6:91_2_1_2_1_1_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:91_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E12:91_2_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:91_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E7:91_2_1_2_1_1_2_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E14:91_2_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_2_1_1_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="finseq_3.html#E6:91_2_1_2_1_1_2_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="finseq_3.html#E12:91_2_1_2_1_1_2_1"><i><font color="Green">E78</font></i></a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E10:91_2_1_2_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></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="E1:91_2_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:91_2_1_2_1_1_2_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E4:91_2_1_2_1_1_2_2"/><i><font color="Green">E76</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> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:91_2_1_2_1_1_2_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E6:91_2_1_2_1_1_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>
<a NAME="E7:91_2_1_2_1_1_2_2"/>
<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> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_2_1_1_2_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:91_2_1_2_1_1_2_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:91_2_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E79</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> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E5:91_2_1_2_1_1_2_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:91_2_1_2_1_1_2_2"/>
not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="finseq_3.html#E7:91_2_1_2_1_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="finseq_3.html#E6:91_2_1_2_1_1_2_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:91_2_1_2_1_1_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E3:91_2_1_2_1_1_2_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="finseq_3.html#E9:91_2_1_2_1_1_2_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:91_2_1_2_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:91_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E13:91_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E14:91_2_1_2_1_1_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E10:91_2_1_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E15:91_2_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E13:91_2_1_2_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>;<br/><a NAME="E16:91_2_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E17:91_2_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E18:91_2_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E19:91_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E76</font></i>: 
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E20:91_2_1_2_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E22:91_2_1_2_1_1_2"/><i><font color="Green">E77</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E19:91_2_1_2_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E23:91_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/><a NAME="E24:91_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</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>1</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E79</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="finseq_3.html#E24:91_2_1_2_1_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/><a NAME="E26:91_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E27:91_2_1_2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E25:91_2_1_2_1_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="card_2.html#T53">CARD_2:53</a></i>;<br/><a NAME="E28:91_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/><a NAME="E29:91_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E30:91_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="finseq_3.html#E10:91_2_1_2_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E31:91_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T63">CARD_2:63</a></i>;<br/><a NAME="E32:91_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></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="E1:91_2_1_2_1_1_2_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:91_2_1_2_1_1_2_4"/><b>then </b><i><font color="Green">E83</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> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E4:91_2_1_2_1_1_2_4"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E4:91_2_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="finseq_3.html#E2:91_2_1_2_1_1_2_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:91_2_1_2_1_1_2_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E2:91_2_1_2_1_1_2_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:91_2_1_2_1_1_2_4"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_2_1_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_2_1_2_1_1_2_4_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:91_2_1_2_1_1_2_4_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:91_2_1_2_1_1_2_4_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:91_2_1_2_1_1_2_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="finseq_3.html#E4:91_2_1_2_1_1_2_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:91_2_1_2_1_1_2_4_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:91_2_1_2_1_1_2_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E6:91_2_1_2_1_1_2_4"><i><font color="Green">E85</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:91_2_1_2_1_1_2"/><b>then </b>
<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>6</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E7:91"><i><font color="Green">E65</font></i></a>, <a class="txt" href="finseq_3.html#E28:91_2_1_2_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="finseq_3.html#E31:91_2_1_2_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E34:91_2_1_2_1_1_2"/><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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T66" target="_self">Th66</a></i>;<br/><a NAME="E35:91_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</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>6</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E22:91_2_1_2_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E36:91_2_1_2_1_1_2"/><b>then </b>
<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"><font color="Maroon">c<sub>6</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E37:91_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="finseq_3.html#E2:91_2_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="finseq_3.html#E8:91_2_1_2_1_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="finseq_3.html#E9:91_2_1_2_1_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:91_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:91"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div>
