<?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="firebrick">::  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   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="E1:15_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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">IT</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="E2:15_1_1_1"/>
 ex <font color="Olive" title="b1">IT</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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="E2:15_1_1_1_1"/><span class="lab"><font color="Green" title="E9">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="E3:15_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="#E1:15"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="integra1.html#T6" target="_self" title="INTEGRA1:th.6">Th8</a></span>;<br/>
<a NAME="E4:15_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#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></span>;<br/>
<a NAME="E5:15_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c4">I</font> is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="#E2:15_1_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="measure5.html#T14" title="MEASURE5:th.14">MEASURE5:14</a></span>;<br/>
<a NAME="E6:15_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">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#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/>
<a NAME="E7:15_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">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="#E2:15_1_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E5:15_1_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th6</a></span>;<br/>
<a NAME="E8:15_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="#E2:15_1_1_1_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="txt" href="#E5:15_1_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E6:15_1_1_1_1"><span class="lab"><font color="Green" title="E11">A4</font></span></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:15_1_1_1_1"/>
 ex <font color="Olive" title="b1">IT</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="#E5:15_1_1_1_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="txt" href="#E7:15_1_1_1_1"><span class="lab"><font color="Green" title="E12">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:15_1_1"/><span class="lab"><font color="Green" title="E9">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="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="E3:15_1_1"/>
 ex <font color="Olive" title="b1">IT</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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="#E1:15"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="integra1.html#T7" target="_self" title="INTEGRA1:th.7">Th9</a></span>;<br/>
<a NAME="E2:15_1_1_2"/><span class="lab"><font color="Green" title="E10">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#T6" title="XREAL_1:th.6">XREAL_1:6</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="E4:15_1_1_2"/><span class="lab"><font color="Green" title="E11">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="E5:15_1_1_2"/><span class="lab"><font color="Green" title="E12">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="E7:15_1_1_2"/><span class="lab"><font color="Green" title="E13">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="E8:15_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="#E1:15"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="integra1.html#T7" target="_self" title="INTEGRA1:th.7">Th9</a></span>;<br/>
<a NAME="E9:15_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="#E1:15"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E2:15_1_1_2"><span class="lab"><font color="Green" title="E10">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="E10:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A11</font></span>: 
<font color="Maroon" title="c7">I</font> is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="#E4:15_1_1_2"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E5:15_1_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E7:15_1_1_2"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="measure5.html#T14" title="MEASURE5:th.14">MEASURE5:14</a></span>;<br/>
<a NAME="E11:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">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#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/>
<a NAME="E12:15_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">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="#E5:15_1_1_2"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E7:15_1_1_2"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E10:15_1_1_2"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th6</a></span>;<br/>
<a NAME="E13:15_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="#E4:15_1_1_2"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="txt" href="#E7:15_1_1_2"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E10:15_1_1_2"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E11:15_1_1_2"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:15_1_1_2"/>
 ex <font color="Olive" title="b1">IT</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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="#E10:15_1_1_2"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E12:15_1_1_2"><span class="lab"><font color="Green" title="E16">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div>
