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

<b>let </b><font color="Maroon">D</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">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>









<b>set </b><font color="Maroon">dx</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">drx</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">frx</font> =  <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">fx</font> =  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font>;<br/>
<b>assume </b><a NAME="E1:98"/><i><font color="Green">E30</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">d</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">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">dx</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<i><font color="Green">E31</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></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">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<font color="Maroon">dx</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E4:98"/><b>then </b><i><font color="Green">E32</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">dx</font> &amp;  <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">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">dx</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:98"/><i><font color="Green">E33</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span> &amp;  <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">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:98"/>
(  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font>,<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp;  <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  )
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E7:98"/><b>then </b><i><font color="Green">E34</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">F</font>,<font color="Maroon">X</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">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/>
<a NAME="E8:98"/><i><font color="Green">E35</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font> &amp; <font color="Maroon">dx</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E9:98"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">d</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">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:98"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E11:98"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</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">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:98"/>
 <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">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E13:98"/><b>then </b>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E14:98"/><b>then </b>
(  <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">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 0 <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">F</font>,<font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E15:98"/><b>then </b>
0 <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">F</font>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:98"/><b>then </b><i><font color="Green">E99</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">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:98"/>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</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">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:98"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">n</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">F</font>,<font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E20:98"/><i><font color="Green">E101</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</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">F</font>,<font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>thus </b><a NAME="E21:98"/>
( <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</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> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> implies <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:98_2"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</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> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E2:98_2"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E3:98_2"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">X</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">d1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E5:98_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dx</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</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">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">X</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E6:98_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</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">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E7:98_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">d1</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">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, <a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>
;<br/>
<a NAME="E9:98_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:98_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">m</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E12:98_2"/><i><font color="Green">E122</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E13:98_2"/>
<font color="Maroon">n</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">F</font>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E14:98_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, , <a class="ref" href="rfunct_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_2_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1_1"><b>case </b></a><a NAME="E1:98_2_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:98_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="rfunct_3.html#T18" target="_self">Th18</a>, <a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>;<br/><a NAME="E3:98_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:98_2_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_2_2_1_2"><b>case </b></a><a NAME="E1:98_2_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:98_2_2_1_2"/><b>then </b>
<font color="Maroon">m</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:98_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , , <a class="ref" href="rfunct_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><a NAME="E4:98_2_2_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E5:98_2_2_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rfunct_3.html#T20" target="_self">Th20</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E16:98_2"/>
contradiction
 ;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E22:98"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</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">F</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E23:98"/><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</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> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E24:98"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</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> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</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">d1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E26:98"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">d1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</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> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E27:98"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">d1</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">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, <a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<i><font color="Green">E163</font></i>: <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</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> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#T17" target="_self">Th17</a>, <a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>
;<br/>
<a NAME="E29:98"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E30:98"/><b>then </b>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</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">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E32:98"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">m</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">F</font>,<font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E33:98"/><i><font color="Green">E165</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</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">F</font>,<font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E34:98"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E35:98"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E37:98"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">n1</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E38:98"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T20" target="_self">Th20</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E39:98"/><b>then </b>
<font color="Maroon">n1</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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="rfunct_3.html#T20" target="_self">Th20</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E40:98"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , , , <a class="ref" href="rfunct_3.html#T20" target="_self">Th20</a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/>
<a NAME="E41:98"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:98_4_1"/>
(  <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">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or  <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">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_1_1"><b>case </b></a><a NAME="E1:98_4_1_1"/>
 <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">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:98_4_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="rfunct_3.html#T18" target="_self">Th18</a>, <a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="98_4_1_2"><b>case </b></a><a NAME="E1:98_4_1_2"/>
 <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">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:98_4_1_2"/><b>then </b>
<font color="Maroon">m</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">F</font>,<font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:98_4_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="rfunct_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rfinseq.html#T32">RFINSEQ:32</a></i>;<br/><b>hence </b><a NAME="E4:98_4_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="rfunct_3.html#T21" target="_self">Th21</a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E43:98"/>
contradiction
 ;<br/>


</div>
