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

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">D</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E32</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/> <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E3:4"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/> <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:4_1"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:4"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/> <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:4_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:4_2"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:4_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:4"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E6:4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:4_3"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E3:4_3"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/>
<a NAME="E4:4_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:4_3"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:4_3"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:4"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E8:4"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E9:4"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> <a href="real_1.html#K3">+</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>hence </b><a NAME="E11:4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>


</div>
