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

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

<b>assume </b><a NAME="E1:96"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>2</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>10</sub></font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:96"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>10</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>

<a NAME="E3:96"/><i><font color="Green">E65</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:96"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E5:96"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E6:96"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font>, <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>, <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E7:96"/><b>then </b><i><font color="Green">E68</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/>
<a NAME="E8:96"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<a NAME="E1:96_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E2:96_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:96"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfunct_3.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E10:96"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:96"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<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">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E13:96"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E7:96"><i><font color="Green">E68</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E14:96"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E13:96"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E15:96"/><i><font color="Green">E73</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:96"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rfunct_3.html#E13:96"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E16:96"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:96_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
  <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="96_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</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="E1:96_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:96_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E15:96"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:96_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>15</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>11</sub></font></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="E4:96_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:96"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rfunct_3.html#E2:96_2_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:96_2_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E3:96_2_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>15</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="E2:96_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:96_2"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>15</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>11</sub></font></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="E4:96_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>15</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="rfunct_3.html#E3:96"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:96_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:96_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:96_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E15:96"><i><font color="Green">E73</font></i></a>, <a class="txt" href="rfunct_3.html#E4:96_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<i><font color="Green">E75</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K5">-</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>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E15:96"><i><font color="Green">E73</font></i></a>, <a class="txt" href="rfunct_3.html#E16:96"><i><font color="Green">E74</font></i></a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>

;<br/>
<a NAME="E19:96"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E20:96"/><i><font color="Green">E77</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E15:96"><i><font color="Green">E73</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<i><font color="Green">E78</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T52">XBOOLE_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E20:96"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>then </b><i><font color="Green">E79</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:96"><i><font color="Green">E65</font></i></a>, <a class="txt" href="rfunct_3.html#E15:96"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E20:96"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:96_6"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> )
 ;<br/><a NAME="E2:96_6"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">c<sub>16</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>11</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E16:96"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_6_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>17</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>11</sub></font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="rfunct_3.html#E22:96"><i><font color="Green">E79</font></i></a>, <a class="txt" href="rfunct_3.html#E1:96_6"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_1_1_1"><b>case </b></a><a NAME="E1:96_6_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:96_6_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E2:96_6"><i><font color="Green">E81</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_1_1_2"><b>case </b></a><a NAME="E1:96_6_1_1_2"/>
<font color="Maroon">c<sub>17</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>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:96_6_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:96_6"><i><font color="Green">E81</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:96_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E2:96_6"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:96_6_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E2:96_6"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:96_6_1_1_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E7:96"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E7:96_6_1_1_2"/><i><font color="Green">E84</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E6:96_6_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_6_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_1_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="E1:96_6_1_1_2_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:96_6_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:96_6_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E13:96"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rfunct_3.html#E2:96_6_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="rfunct_3.html#E6:96_6_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_6_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:96_6_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:96_6_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E7:96_6_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:96_6_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E10:96"><i><font color="Green">E70</font></i></a>, <a class="txt" href="rfunct_3.html#E14:96"><i><font color="Green">E72</font></i></a>, <a class="txt" href="rfunct_3.html#E2:96_6_1_1_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="rfunct_3.html#E6:96_6_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:96_6_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:96_6"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E24:96"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:96"><i><font color="Green">E63</font></i></a>, <a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E18:96"><i><font color="Green">E75</font></i></a>, <a class="txt" href="rfunct_3.html#E19:96"><i><font color="Green">E76</font></i></a></i>;<br/>
<i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></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>11</sub></font></span><a href="tarski.html#K1">}</a></span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T41">XBOOLE_1:41</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E20:96"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E26:96"/><i><font color="Green">E82</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E5:96"><i><font color="Green">E66</font></i></a>, <a class="txt" href="rfunct_3.html#E22:96"><i><font color="Green">E79</font></i></a>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<a NAME="E27:96"/>
<font color="Maroon">c<sub>11</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>11</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E28:96"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E21:96"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E29:96"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E15:96"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:96"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E25:96"><i><font color="Green">E81</font></i></a>, <a class="txt" href="rfunct_3.html#E26:96"><i><font color="Green">E82</font></i></a>, <a class="ref" href="rfunct_3.html#T69" target="_self">Th69</a></i>;<br/>
<a NAME="E31:96"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq.html#T15">RFINSEQ:15</a></i>;<br/>
<a NAME="E32:96"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E30:96"><i><font color="Green">E83</font></i></a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<a NAME="E33:96"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E26:96"><i><font color="Green">E82</font></i></a>, <a class="ref" href="rfunct_3.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E34:96"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>, <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E32:96"><i><font color="Green">E84</font></i></a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<a NAME="E35:96"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> is <a href="rfinseq.html#V1">non-increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfinseq.html#D4">RFINSEQ:def 4</a><br/>

<b>assume </b><a NAME="E1:96_8"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E2:96_8"/><i><font color="Green">E87</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E3:96_8"/><i><font color="Green">E88</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:96_8"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:96_8"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> 0,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T4">FINSEQ_2:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="finseq_2.html#T5">FINSEQ_2:5</a></i>;<br/>
<a NAME="E6:96_8"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96_8"><i><font color="Green">E87</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E7:96_8"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
;<br/>
<a NAME="E8:96_8"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:96_8"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:96_8"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:96_8"><i><font color="Green">E88</font></i></a>, <a class="txt" href="rfunct_3.html#E7:96_8"><i><font color="Green">E89</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E10:96_8"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:96_8"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96_8"><i><font color="Green">E87</font></i></a>, <a class="txt" href="rfunct_3.html#E3:96_8"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/>
<a NAME="E12:96_8"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E10:96_8"><i><font color="Green">E91</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E14:96_8"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E7:96"><i><font color="Green">E68</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E15:96_8"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E14:96_8"><i><font color="Green">E93</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E16:96_8"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E13:96"><i><font color="Green">E71</font></i></a>, <a class="txt" href="rfunct_3.html#E14:96_8"><i><font color="Green">E93</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_8_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_8_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:96_8_2_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_8_2_1_1"><b>suppose </b></a><a NAME="E1:96_8_2_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:96_8_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E11:96_8"><i><font color="Green">E92</font></i></a>, <a class="txt" href="rfunct_3.html#E15:96_8"><i><font color="Green">E94</font></i></a>, <a class="txt" href="rfunct_3.html#E16:96_8"><i><font color="Green">E95</font></i></a>, <a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="96_8_2_1_2"><b>suppose </b></a><a NAME="E1:96_8_2_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:96_8_2_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E3:96_8"><i><font color="Green">E88</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:96_8_2_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:96_8"><i><font color="Green">E87</font></i></a>, <a class="txt" href="rfunct_3.html#E3:96_8"><i><font color="Green">E88</font></i></a>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>;<br/><a NAME="E4:96_8_2_1_2"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:96_8_2_1_2"><i><font color="Green">E96</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:96_8_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:96_8_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E8:96_8"><i><font color="Green">E90</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E7:96_8_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E10:96_8"><i><font color="Green">E91</font></i></a>, <a class="txt" href="rfunct_3.html#E11:96_8"><i><font color="Green">E92</font></i></a>, <a class="txt" href="rfunct_3.html#E3:96_8_2_1_2"><i><font color="Green">E97</font></i></a>, <a class="ref" href="rfinseq.html#D4">RFINSEQ:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:96_8"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E36:96"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E34:96"><i><font color="Green">E85</font></i></a>, <a class="ref" href="rfinseq.html#T36">RFINSEQ:36</a></i>;<br/>
<a NAME="E37:96"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E24:96"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E38:96"/>
<font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:96"><i><font color="Green">E64</font></i></a>, <a class="txt" href="rfunct_3.html#E13:96"><i><font color="Green">E71</font></i></a>, <a class="ref" href="rfunct_3.html#T69" target="_self">Th69</a></i>;<br/>
<a NAME="E39:96"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>, <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E6:96"><i><font color="Green">E67</font></i></a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<a NAME="E40:96"/><i><font color="Green">E88</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E9:96"><i><font color="Green">E69</font></i></a>, <a class="txt" href="rfunct_3.html#E14:96"><i><font color="Green">E72</font></i></a>, <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/>
<a NAME="E41:96"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E39:96"><i><font color="Green">E87</font></i></a>, <a class="ref" href="rfinseq.html#T14">RFINSEQ:14</a></i>;<br/>
<a NAME="E42:96"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rfinseq.html#V1">non-increasing</a>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T33">RFINSEQ:33</a></i>;<br/>
<b>hence </b><a NAME="E43:96"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E37:96"><i><font color="Green">E86</font></i></a>, <a class="txt" href="rfunct_3.html#E40:96"><i><font color="Green">E88</font></i></a>, <a class="txt" href="rfunct_3.html#E41:96"><i><font color="Green">E89</font></i></a>, <a class="ref" href="rfinseq.html#T36">RFINSEQ:36</a></i>;<br/>


</div>
