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

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  ex <font color="Olive">b<sub>1</sub></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:20_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )</span><br/><span class="kw">thus </span><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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">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> <span class="kw"> such that </span><br/><a NAME="E2:20_1_1_1_1"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c4">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>
<a NAME="E3:20_1_1_1_1"/>
<font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="integra1.html#T8" target="_self" title="INTEGRA1:th.8">Th8</a></span>;<br/>
<a NAME="E4:20_1_1_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></span>;<br/>
<a NAME="E5:20_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c4">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="integra1.html#D1" target="_self" title="INTEGRA1:def.1">Def1</a></span>;<br/>
<a NAME="E6:20_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c4">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c4">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c4">I</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></span>;<br/>
<a NAME="E7:20_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A5</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E5:20_1_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="integra1.html#T6" target="_self" title="INTEGRA1:th.6">Th6</a></span>;<br/>
<a NAME="E8:20_1_1_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E5:20_1_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E6:20_1_1_1_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="ref" href="integra1.html#T6" target="_self" title="INTEGRA1:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:20_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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E7:20_1_1_1_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E13">A6</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )</span><br/>

<span class="kw">thus </span><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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c4">j</font> = <font color="Maroon" title="c3">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="integra1.html#T9" target="_self" title="INTEGRA1:th.9">Th9</a></span>;<br/>
<a NAME="E2:20_1_1_2"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
<font color="Maroon" title="c3">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="c3">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#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:20_1_1_2"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:20_1_1_2"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">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> <span class="kw"> such that </span><br/><a NAME="E7:20_1_1_2"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<font color="Maroon" title="c7">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="E8:20_1_1_2"/>
<font color="Maroon" title="c3">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="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E13">A6</font></span></a>, <a class="ref" href="integra1.html#T9" target="_self" title="INTEGRA1:th.9">Th9</a></span>;<br/>
<a NAME="E9:20_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:20_1_1_2"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E10:20_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<font color="Maroon" title="c7">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E5:20_1_1_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E7:20_1_1_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="integra1.html#D1" target="_self" title="INTEGRA1:def.1">Def1</a></span>;<br/>
<a NAME="E11:20_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A12</font></span>: 
<font color="Maroon" title="c7">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c7">I</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c7">I</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></span>;<br/>
<a NAME="E12:20_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A13</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c7">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E7:20_1_1_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E10:20_1_1_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="integra1.html#T6" target="_self" title="INTEGRA1:th.6">Th6</a></span>;<br/>
<a NAME="E13:20_1_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c7">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E7:20_1_1_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E10:20_1_1_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E11:20_1_1_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="integra1.html#T6" target="_self" title="INTEGRA1:th.6">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:20_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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Olive" title="b1">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20_1_1_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E12:20_1_1_2"><span class="lab"><font color="Green" title="E20">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
