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

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



<b>assume </b><a NAME="E1:49"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 ;<br/>

<b>thus </b><a NAME="E2:49"/>
( <font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">p</font> <a href="finseq_4.html#K6" target="_self">-|</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K6" target="_self">-|</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="txt" href="finseq_4.html#E1:49"><i><font color="Green">E32</font></i></a>, </i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="finseq_4.html#E1:49_1"><i><font color="Green">E33</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E3:49_1"/>
<font color="Maroon">p</font> <a href="finseq_4.html#K6" target="_self">-|</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:49"/>
<font color="Maroon">p</font> <a href="finseq_4.html#K6" target="_self">-|</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E4:49"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K6" target="_self">-|</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_4.html#K6" target="_self">-|</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="txt" href="finseq_4.html#E1:49"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E5:49"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 1
 
;<br/>
<b>hence </b><a NAME="E6:49"/>
<font color="Maroon">x</font> <a href="finseq_4.html#K5" target="_self">..</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


</div>
