<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="integra1.html#M2" title="INTEGRA1:mode.2">Division</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">D1</font>, <font color="Olive" title="b3">D2</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b3">D2</font> holds <br/> <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Olive" title="b3">D2</font>,<font color="Olive" title="b2">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">D2</font></span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="integra1.html#M2" title="INTEGRA1:mode.2">Division</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D1</font>, <font color="Olive" title="b2">D2</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b2">D2</font> holds <br/> <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Olive" title="b2">D2</font>,<font color="Olive" title="b1">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">D2</font></span><br/><b>let </b><font color="Maroon" title="c3">D1</font>, <font color="Maroon" title="c4">D2</font> be    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c4">D2</font> implies  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font> )</span><br/>





<b>assume </b><a NAME="E1:73"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c3">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c4">D2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font></span><br/>

<b>assume </b><a NAME="E2:73"/><i><font color="Green" title="E55">A2</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:73"/><i><font color="Green" title="E56">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</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="c3">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<a NAME="E1:73_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E2:73_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>hence </b><a NAME="E3:73_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</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="c3">D1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:73"/><b>then </b>
<font color="Maroon" title="c3">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:73"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<a NAME="E5:73"/><b>then </b><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</a></i>;<br/>
<a NAME="E6:73"/><i><font color="Green" title="E58">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</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="c4">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>proof </b></a><div class="add">

<a NAME="E1:73_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E2:73_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>hence </b><a NAME="E3:73_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</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="c4">D2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:73"/><i><font color="Green" title="E59">A6</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <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="c4">D2</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:73"><i><font color="Green" title="E54">A1</font></i></a>, <a class="txt" href="integra1.html#E3:73"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<a NAME="E8:73"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<a NAME="E1:73_3"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E7:73"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:73_3"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:73_3"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:73"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:73"/><b>then </b>
<font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c3">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E6:73"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="integra1.html#E7:73"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:73"/>
contradiction
 <b>by </b><i><a class="txt" href="integra1.html#E5:73"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
