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

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">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">D</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>





<b>assume </b><a NAME="E1:5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:5"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E3:5"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E4:5"/><b>then </b><i><font color="Green">E39</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1"><b>suppose </b></a><a NAME="E1:5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">j</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="E3:5_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E4:5_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</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="5_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:5_1_1_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:5_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra1.html#T4">INTEGRA1:4</a></i>;<br/><b>thus </b><a NAME="E5:5_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 <b>by </b><i>, , , <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:5_1_1_1_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:5_1_1_1_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra1.html#T4">INTEGRA1:4</a></i>;<br/><b>thus </b><a NAME="E5:5_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font>
 <b>by </b><i>, , , <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E5:5_1_1"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">j</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2"><b>suppose </b></a><a NAME="E1:5_1_2"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> );<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> );<br/><a NAME="E2:5_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E3:5_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_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="E1:5_1_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</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="5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:5_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></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><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/><a NAME="E5:5_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:5_1_2_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1_2_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:5_1_2_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:5_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i/>;<br/><b>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="E8:5_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><a NAME="E9:5_1_2"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E10:5_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 <b>by </b><i>, </i>;<br/><a NAME="E11:5_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>consider </b><font color="Maroon">j</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="E13:5_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">j</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i><a NAME="E14:5_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1_2_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_2_3_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:5_1_2_3_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:5_1_2_3_1_1"/><i><font color="Green">E58</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:5_1_2_3_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E4:5_1_2_3_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_3_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_2_3_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</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="5_1_2_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_2_3_1_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_2_3_1_1_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_2_3_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_2_3_1_1_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:5_1_2_3_1_1"/>
<font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E6:5_1_2_3_1_1"/><b>then </b>
<font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:5_1_2_3_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E8:5_1_2_3_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E9:5_1_2_3_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:5_1_2_3_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:5_1_2_3_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_3_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1_2_3_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span>
 ;<br/>

<a NAME="E2:5_1_2_3_1_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_3_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_3_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_2_3_1_2_1_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</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="5_1_2_3_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_2_3_1_2_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_2_3_1_2_1_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_3_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_2_3_1_2_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_2_3_1_2_1_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:5_1_2_3_1_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_3_1_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:5_1_2_3_1_2_1_2"/><i><font color="Green">E65</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E2:5_1_2_3_1_2_1_2"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:5_1_2_3_1_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</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> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:5_1_2_3_1_2_1_2"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<b>thus </b><a NAME="E5:5_1_2_3_1_2_1_2"/>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</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="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>

;<br/>
<a NAME="E5:5_1_2_3_1_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:5_1_2_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:5_1_2_3_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E15:5_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> )
 <b>by </b><i>, , <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><b>hence </b><a NAME="E16:5_1_2"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">j</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
