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

<b>let </b><font color="Maroon">f1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:16"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D6">FINSEQ_4:def 6</a></i>;<br/>
<a NAME="E5:16"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E6:16"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:16"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:16"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:16"/>
<font color="Maroon">p</font> <a href="finseq_4.html#K5">..</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T31">FINSEQ_4:31</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:16"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</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">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E13:16"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:16"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f2</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:16"/>
<span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="finseq_4.html#K6">-|</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="finseq_6.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="txt" href="finseq_6.html#E2"><i><font color="Green">Lemma18</font></i></a>, , <a class="ref" href="finseq_4.html#D6">FINSEQ_4:def 6</a></i>;<br/>


</div>
