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

<b>let </b><font color="Maroon">d</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font></span>)</span>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K5" target="_self">cells</a> 0,<font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">l</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font>, <font color="Maroon">r</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E3:54_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">r</font> &amp; (  ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) or ( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T1" target="_self">Th1</a></i>;<br/><b>consider </b><font color="Maroon">X</font> being   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span><b> such that </b><br/><a NAME="E5:54_1"/><i><font color="Green">E51</font></i>: 
(  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">l'</font> = <font color="Maroon">l</font>, <font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><a NAME="E7:54_1"/>
<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="card_2.html#T59">CARD_2:59</a></i>;<br/><a NAME="E8:54_1"/><b>then </b><i><font color="Green">E52</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds <br/> ( <font color="Maroon">l'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> )
 <b>by </b><i/>;<br/><a NAME="E9:54_1"/><b>then </b>
<font color="Maroon">l'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E10:54_1"/>
 ex <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Olive">x</font>,<font color="Olive">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Olive">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) )
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">x</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><a NAME="E2:54"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) )
 ;<br/>

<a NAME="E3:54"/>
 ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> st <br/>(  <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">X</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E2:54_2"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>  holds <br/>( ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> is    <a href="chain_1.html#M2" target="_self">Gap</a> of <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) or ( not <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) ) )
 <b>by </b><i>, <a class="ref" href="card_2.html#T19">CARD_2:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K5" target="_self">cells</a> 0,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#T1" target="_self">Th1</a></i>;<br/>


</div>
