<?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" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><b>thus </b><a NAME="E2:20_1_1_1"/>
 ex <font color="Olive" title="b1">IT</font> being <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">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" title="E14">A2</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:20"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="integra1.html#T8" target="_self" title="INTEGRA1:th.8">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">I</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:20_1_1_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>
<a NAME="E4:20_1_1_1_1"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">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" title="c1">A</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="integra1.html#T3" target="_self" title="INTEGRA1:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:20_1_1_1_1_1"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:20_1_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">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" title="E16">A4</font></i>: 
<font color="Maroon" title="c5">I</font> is  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E3:20_1_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="integra1.html#D1" target="_self" title="INTEGRA1:def.1">Def1</a></i>;<br/>
<a NAME="E6:20_1_1_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">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" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c5">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c5">I</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:20_1_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E2:20_1_1_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c5">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E3:20_1_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="integra1.html#E5:20_1_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="integra1.html#T6" target="_self" title="INTEGRA1:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20_1_1_1_1"/>
 ex <font color="Olive" title="b1">IT</font> being <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E5:20_1_1_1_1"><i><font color="Green" title="E16">A4</font></i></a></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" title="E14">A5</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<b>thus </b><a NAME="E3:20_1_1"/>
 ex <font color="Olive" title="b1">IT</font> being <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">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" title="E15">A6</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:20"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="integra1.html#E2:20_1_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="integra1.html#T9" target="_self" title="INTEGRA1:th.9">Th9</a></i>;<br/>
<a NAME="E2:20_1_1_2"/><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">b1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:20_1_1_2"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">j</font> = <font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="integra1.html#E1:20"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="integra1.html#E2:20_1_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="integra1.html#T9" target="_self" title="INTEGRA1:th.9">Th9</a></i>;<br/>
<a NAME="E6:20_1_1_2"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:20"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="integra1.html#E1:20_1_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="integra1.html#E2:20_1_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">I</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E8:20_1_1_2"/><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c5">a1</font>,<font color="Maroon" title="c6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>
<a NAME="E9:20_1_1_2"/><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c8">I</font> is  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E4:20_1_1_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="integra1.html#E6:20_1_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="integra1.html#E8:20_1_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="integra1.html#D1" target="_self" title="INTEGRA1:def.1">Def1</a></i>;<br/>
<a NAME="E10:20_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c8">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c8">I</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<a NAME="E11:20_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c8">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E4:20_1_1_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="integra1.html#E8:20_1_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="integra1.html#E9:20_1_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="integra1.html#T6" target="_self" title="INTEGRA1:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E12:20_1_1_2"/>
 ex <font color="Olive" title="b1">IT</font> being <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E9:20_1_1_2"><i><font color="Green" title="E20">A11</font></i></a></i>;<br/>


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


</div>
