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

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1" target="_self">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">D1</font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">D2</font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font>;<br/>





<b>consider </b><font color="Maroon">D3</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:77"/><i><font color="Green">E49</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D3</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span></span>)</span> &amp; <font color="Maroon">D3</font> is <a href="goboard1.html#V1">increasing</a> )
 <b>by </b><i><a class="ref" href="goboard2.html#T21">GOBOARD2:21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D3</font> = <font color="Maroon">D3</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="goboard1.html#V1">increasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<a NAME="E4:77"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:77"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:77"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E7:77"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E8:77"/>
<font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:77_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_1_1"/>
( <font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> or <font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:77_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:77_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E4:77_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:77_1_1_1_1"/><b>then </b>
<font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D3</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:77_1_1_1_1"/><b>then </b>
<font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E7:77_1_1_1_1"/><b>then </b>
<font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>;<br/><a NAME="E8:77_1_1_1_1"/><b>then </b>
<font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : (  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:77_1_1_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 ;<br/><b>thus </b><a NAME="E11:77_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="integra1.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><a NAME="E2:77_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:77_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E4:77_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:77_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E6:77_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E7:77_1_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:77_1_1_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><a NAME="E9:77_1_1_1_2"/>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><a NAME="E10:77_1_1_1_2"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:77_1_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D3</font> &amp; <font color="Maroon">D3</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E1:77_1_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra1.html#E8:77_1_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E13:77_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E12:77_1_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:77_1_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E15:77_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="integra1.html#T6" target="_self">Th6</a>, , <a class="txt" href="integra1.html#E12:77_1_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:77_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:77"/><b>then </b>
<font color="Maroon">D3</font> is    <a href="integra1.html#M1" target="_self">DivisionPoint</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">D3</font> = <font color="Maroon">D3</font> as    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E11:77"/>
<font color="Maroon">D1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jordan7.html#T17">JORDAN7:17</a></i>;<br/>
<a NAME="E12:77"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E13:77"/>
<font color="Maroon">D2</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jordan7.html#T17">JORDAN7:17</a></i>;<br/>
<a NAME="E14:77"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E15:77"/><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/>
<a NAME="E16:77"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font>
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E17:77"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/>
<a NAME="E18:77"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D3</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T6" target="_self">Th6</a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<b>take </b>
<font color="Maroon">D3</font>
;<br/>

<b>thus </b><a NAME="E19:77"/>
( <font color="Maroon">D1</font> <a href="integra1.html#R4" target="_self">&lt;=</a> <font color="Maroon">D3</font> &amp; <font color="Maroon">D2</font> <a href="integra1.html#R4" target="_self">&lt;=</a> <font color="Maroon">D3</font> )
 <b>by </b><i>, , , , , </i>;<br/>


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