<?xml version="1.0"?>
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<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">D1</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>, <font color="Maroon">D2</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<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> ;<br/><b>let </b><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><a NAME="E1:27"/><i><font color="Green">E32</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">D1</font>
 ;<br/>

<b>assume </b><a NAME="E2:27"/><i><font color="Green">E39</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">D1</font>
 ;<br/>

<b>assume </b><a NAME="E3:27"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<b>assume </b><a NAME="E4:27"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D2</font>
 ;<br/>

<b>assume </b><a NAME="E5:27"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1
 ;<br/>

<b>set </b><font color="Maroon">MD1</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">MD2</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>;<br/>
<a NAME="E6:27"/><i><font color="Green">E43</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="txt" href="integra3.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E7:27"/><i><font color="Green">E44</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="txt" href="integra3.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E8:27"/>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E9:27"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="integra3.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being  <a href="integra1.html#V1">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="E11:27"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></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">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> is    <a href="integra1.html#M1">DivisionPoint</a> of <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="integra3.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="integra3.html#E4"><i><font color="Green">Lemma35</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">MD2</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">B</font> <b>by </b><i>, <a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<a NAME="E13:27"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra3.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E14:27"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E15:27"/><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="integra3.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E52</font></i>: <font color="Maroon">MD2</font> <a href="goboard1.html#K1">.</a> 1 = 
<font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="integra3.html#T1" target="_self">Th1</a>, , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="txt" href="integra3.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>
;<br/>
<a NAME="E17:27"/>
<font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E18:27"/><b>then </b><i><font color="Green">E53</font></i>: 
<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 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E19:27"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then </b><i><font color="Green">E55</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="txt" href="integra3.html#E4:27"><i><font color="Green">E41</font></i></a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>

;<br/>
<b>then consider </b><font color="Maroon">C</font> being  <a href="integra1.html#V1">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="E22:27"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">C</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</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">D1</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</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 &amp;  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> is    <a href="integra1.html#M1">DivisionPoint</a> of <font color="Maroon">C</font> )
 <b>by </b><i>, , <a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="integra3.html#E3:27"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<i><font color="Green">E57</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</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">D1</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
<font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="txt" href="integra3.html#E4:27"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra3.html#E5:27"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra3.html#E7:27"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>

;<br/>
<i><font color="Green">E58</font></i>: <font color="Maroon">MD2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font></span>)</span> = 
<font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra3.html#T1" target="_self">Th1</a>, , , <a class="txt" href="integra3.html#E2:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="txt" href="integra3.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>
;<br/>
<i><font color="Green">E59</font></i>: <font color="Maroon">B</font> = 
<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="integra1.html#T5">INTEGRA1:5</a></i>
<br/>.= 
<font color="Maroon">C</font>
<b>by </b><i>, <a class="txt" href="integra3.html#E7:27"><i><font color="Green">E44</font></i></a>, <a class="txt" href="integra3.html#E9:27"><i><font color="Green">E45</font></i></a>, <a class="ref" href="integra3.html#T2" target="_self">Th2</a>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">MD1</font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">B</font> <b>by </b><i><a class="txt" href="integra3.html#E7:27"><i><font color="Green">E44</font></i></a>, <a class="ref" href="integra1.html#D3">INTEGRA1:def 3</a></i>;<br/>
<a NAME="E27:27"/><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_6"><b>proof </b></a><div class="add">

<a NAME="E1:27_6"/><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T28">JORDAN3:28</a></i>;<br/>
<a NAME="E2:27_6"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="integra1.html#D20">INTEGRA1:def 20</a></i>;<br/>
<a NAME="E3:27_6"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:27_6"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:27_6"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E14:27"><i><font color="Green">E50</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:27_6"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font> &amp; <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E15:27"><i><font color="Green">E51</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E9:27_6"/><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E18:27"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:27_6"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:27_6"/><b>then </b><i><font color="Green">E71</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E12:27_6"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E13:27_6"/><b>then </b>
( <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E18:27"><i><font color="Green">E53</font></i></a>, <a class="txt" href="integra3.html#E19:27"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E20:27"><i><font color="Green">E55</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E14:27_6"/><b>then </b><i><font color="Green">E72</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="integra3.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="integra3.html#E6:27"><i><font color="Green">E43</font></i></a>, <a class="txt" href="integra3.html#E9:27"><i><font color="Green">E45</font></i></a>, <a class="txt" href="integra3.html#E16:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra3.html#E18:27"><i><font color="Green">E53</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E15:27_6"/><b>then </b>
( <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E16:27_6"/><b>then </b><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E17:27_6"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E22:27"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>then consider </b><font color="Maroon">k3</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:27_6"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">k2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k3</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E20:27_6"/>
<font color="Maroon">k3</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E21:27_6"/><b>then </b>
<font color="Maroon">MD2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k3</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T1" target="_self">Th1</a>, <a class="txt" href="integra3.html#E2:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="integra3.html#E23:27"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra3.html#E24:27"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/>
<a NAME="E22:27_6"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">MD2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E24:27"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E23:27_6"/>
<font color="Maroon">k3</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD2</font>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E23:27"><i><font color="Green">E57</font></i></a>, <a class="txt" href="integra3.html#E24:27"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E24:27_6"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E16:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="integra3.html#E25:27"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:27"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_7"><b>proof </b></a><div class="add">

<a NAME="E1:27_7"/>
( <font color="Maroon">MD1</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">MD2</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="ref" href="jordan7.html#T17">JORDAN7:17</a></i>;<br/>
<a NAME="E2:27_7"/><b>then </b>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>;<br/>
<b>hence </b><a NAME="E3:27_7"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font>
 <b>by </b><i>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:27"/><b>then </b>
<font color="Maroon">MD1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">MD2</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D20">INTEGRA1:def 20</a></i>;<br/>
<b>hence </b><a NAME="E30:27"/>
 ex <font color="Olive">B</font> being <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ex <font color="Olive">MD1</font>, <font color="Olive">MD2</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Olive">B</font> st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Olive">B</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">MD2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">MD2</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">MD1</font></span>)</span> &amp; <font color="Olive">MD1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">MD2</font> &amp; <font color="Olive">MD1</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> &amp; <font color="Olive">MD2</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="integra3.html#E7:27"><i><font color="Green">E44</font></i></a>, </i>;<br/>


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