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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>thus </b><a NAME="E2:20_1_1_1"/>
 ex <font color="Olive">IT</font> being <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>  st <br/>(  <a href="pscomp_1.html#K3">inf</a> <font color="Olive">IT</font> <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> <font color="Olive">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">I</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:20_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</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"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>
<a NAME="E4:20_1_1_1_1"/>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_1_1_1"/>
<font color="Maroon">A</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:20_1_1_1_1_1"/>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:20_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">I</font> is  <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> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:20_1_1_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">I</font> <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> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_1_1_2"/>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</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">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">I</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:20_1_1_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">I</font> <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> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20_1_1_1_1"/>
 ex <font color="Olive">IT</font> being <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>  st <br/>(  <a href="pscomp_1.html#K3">inf</a> <font color="Olive">IT</font> <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> <font color="Olive">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:20_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<b>thus </b><a NAME="E3:20_1_1"/>
 ex <font color="Olive">IT</font> being <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>  st <br/>(  <a href="pscomp_1.html#K3">inf</a> <font color="Olive">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Olive">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">i</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">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:20_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:20_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , </i>;<br/>
<a NAME="E6:20_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">I</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E8:20_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>
<a NAME="E9:20_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">I</font> is  <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> 
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E10:20_1_1_2"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R1">=</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">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">I</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:20_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E12:20_1_1_2"/>
 ex <font color="Olive">IT</font> being <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>  st <br/>(  <a href="pscomp_1.html#K3">inf</a> <font color="Olive">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Olive">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/>


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


</div>
