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

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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">j1</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">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>assume </b><a NAME="E1:15"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 ;<br/>

<b>assume </b><a NAME="E2:15"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="seq_4.html#K1">}</a></span> )
 ;<br/>

<a NAME="E3:15"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E4:15"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E5:15"/><b>then </b><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</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>  &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</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">D1</font> )
 
<b>by </b><i>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/>
<a NAME="E6:15"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E7:15"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</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">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E8:15"/><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">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, , , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E9:15"/><b>then </b><i><font color="Green">E45</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">j1</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</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="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:15"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, , , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/>
<a NAME="E11:15"/><b>then </b><i><font color="Green">E50</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</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/>
<a NAME="E12:15"/>
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">k</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="E3:15_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:15_1"/>
<font color="Maroon">k</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> <span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:15_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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">j1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:15_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<a NAME="E7:15_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:15_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</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">j1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <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">j1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E10:15_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, , , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E11:15_1"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E12:15_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="seq_4.html#K1">}</a></span> implies <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="seq_4.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:15_1_1"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:15_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="seq_4.html#K1">}</a></span> implies <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="seq_4.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:15_1_2"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E1:15"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:15_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:15_1_2"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:15_1_2"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 <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#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="integra3.html#T1" target="_self">Th1</a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:15_1"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> implies <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_3"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</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="E3:15_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:15_1_3"/>
<font color="Maroon">m</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">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:15_1_3"/><b>then </b><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</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#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E6:15_1_3"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:15"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:15_1_3"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="ref" href="integra3.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:15_1_3"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<b>hence </b><a NAME="E9:15_1_3"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:15_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 <b>by </b><i>, , , , , <a class="txt" href="integra3.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:15"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E14:15"/>
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">k</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="E3:15_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:15_2"/>
<font color="Maroon">k</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> <span class="p2">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:15_2"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:15_2"/><b>then </b><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<a NAME="E7:15_2"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:15_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</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="E10:15_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">n</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">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E11:15_2"/><i><font color="Green">E73</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</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">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , , <a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E12:15_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:15_2"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, , , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E14:15_2"/><b>then </b>
<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">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E15:15_2"/><b>then </b>
<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">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <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">j1</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E16:15_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">n</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">j1</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#E1:15"><i><font color="Green">E32</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E17:15_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:15_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</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">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E19:15_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<a NAME="E20:15_2"/>
<font color="Maroon">n</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> <span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E21:15_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E22:15_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></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"><i><font color="Green">Lemma35</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="E15:15"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E16:15"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</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">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
