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

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1" target="_self">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">B</font> be  <a href="integra1.html#V1" target="_self">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">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">S1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of <font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">D</font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S</font>;<br/><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> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:66"/><i><font color="Green">E49</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E3:66"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E4:66"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:66"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">B</font>
 ;<br/>

<b>consider </b><font color="Maroon">A1</font> being  <a href="integra1.html#V1" target="_self">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E7:66"/><i><font color="Green">E57</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> 1
 <b>and </b><br/><a NAME="E8:66"/><i><font color="Green">E58</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:66"/><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>and </b><br/><a NAME="E10:66"/><i><font color="Green">E61</font></i>: 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> is    <a href="integra1.html#M1" target="_self">DivisionPoint</a> of <font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#T1" target="_self">Th1</a>, <a class="txt" href="integra1.html#E1:66"><i><font color="Green">E49</font></i></a>, </i>;<br/>
<a NAME="E11:66"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:66"/><b>then </b><i><font color="Green">E65</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:66"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:66"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">j</font> <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="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:66"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:66"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E16:66"/><b>then </b><i><font color="Green">E67</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E17:66"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
;<br/>
<a NAME="E18:66"/><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
;<br/>
<a NAME="E19:66"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<a NAME="E1:66_1"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:66_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:66_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A1</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:66_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : (  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A1</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E4:66_1_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp;  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A1</font> )
 
;<br/>
<a NAME="E5:66_1_1"/><b>then </b>
( <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E1:66"><i><font color="Green">E49</font></i></a>, <a class="ref" href="integra1.html#T4" target="_self">Th4</a>, <a class="txt" href="integra1.html#E2:66"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra1.html#E3:66"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra1.html#E5:66"><i><font color="Green">E56</font></i></a>, <a class="txt" href="integra1.html#E7:66"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra1.html#E8:66"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra1.html#E9:66"><i><font color="Green">E59</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/>
<a NAME="E6:66_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="integra1.html#T2" target="_self">Th2</a>, <a class="ref" href="integra1.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:66_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : (  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">B</font> ) </span>}</span> 
 
;<br/>
<a NAME="E8:66_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">B</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:66_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:66_1"/>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:66"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:66"><i><font color="Green">E55</font></i></a>, </i>;<br/>
<a NAME="E21:66"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E10:66"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E22:66"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:66"><i><font color="Green">E49</font></i></a>, <a class="txt" href="integra1.html#E3:66"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra1.html#E5:66"><i><font color="Green">E56</font></i></a>, <a class="txt" href="integra1.html#E7:66"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra1.html#E8:66"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/>
<a NAME="E23:66"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> is    <a href="integra1.html#M1" target="_self">DivisionPoint</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T3" target="_self">Th3</a>, <a class="txt" href="integra1.html#E4:66"><i><font color="Green">E55</font></i></a>, <a class="ref" href="integra1.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>hence </b><a NAME="E24:66"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">D</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> is    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">S1</font>
 <b>by </b><i/>;<br/>


</div>
