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

<b>let </b><font color="Maroon" title="c1">A</font> be  <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> ;<br/><b>let </b><font color="Maroon" title="c2">D</font> be    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be    <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> ;<br/>







<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c4">i</font> <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>
 ;<br/>

<b>assume </b><a NAME="E2:19"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c5">j</font> <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>
 ;<br/>

<b>assume </b><a NAME="E3:19"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font>
 ;<br/>

<b>assume </b><a NAME="E4:19"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">C</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> <b> such that </b><br/><a NAME="E6:19"/><i><font color="Green" title="E24">A5</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">DivisionPoint</a> of <font color="Maroon" title="c6">C</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="integra3.html#E2:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="integra3.html#E3:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="integra1.html#T38" title="INTEGRA1:th.38">INTEGRA1:38</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E8:19"/><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> &amp; <font color="Maroon" title="c7">a</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="c6">C</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">C</font> )
 <b>by </b><i><a class="ref" href="integra1.html#T4" title="INTEGRA1:th.4">INTEGRA1:4</a></i>;<br/>
<a NAME="E9:19"/>
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">C</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:19"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="integra3.html#E6:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="integra3.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">B</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">C</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as  <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="ref" href="integra1.html#D1" title="INTEGRA1:def.1">INTEGRA1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">MD</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="integra3.html#E6:19"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<a NAME="E12:19"/>
<font color="Maroon" title="c9">B</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="c9">B</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="c9">B</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" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E13:19"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">C</font> )
 
<b>by </b><i><a class="ref" href="integra1.html#T6" title="INTEGRA1:th.6">INTEGRA1:6</a></i>;<br/>
<a NAME="E14:19"/><i><font color="Green" title="E27">A8</font></i>: 
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font> &amp; <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">C</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E15:19"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c6">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">C</font> )
 
<b>by </b><i><a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c6">C</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:19"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="integra3.html#E6:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="integra3.html#E13:19"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:19"/><b>then </b>
<font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<a NAME="E17:19"/><b>then </b>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:19"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E18:19"/><b>then </b>
<font color="Maroon" title="c10">MD</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">DivisionPoint</a> of <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="integra3.html#E13:19"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E19:19"/>
 ex <font color="Olive" title="b1">B</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/>( <font color="Maroon" title="c3">r</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="Olive" title="b1">B</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 &amp;  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">DivisionPoint</a> of <font color="Olive" title="b1">B</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E6:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="integra3.html#E13:19"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>


</div>
