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

<b>let </b><font color="Maroon" title="c1">d</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">l</font>, <font color="Maroon" title="c3">r</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">d</font>;<br/><b>let </b><font color="Maroon" title="c4">G</font> be    <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c1">d</font>;<br/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:73_1"/>
 <a href="chain_1.html#K4" title="CHAIN_1:func.4">cell</a> <font color="Maroon" title="c2">l</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">infinite-cell</a> <font color="Maroon" title="c4">G</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">l'</font>, <font color="Maroon" title="c6">r'</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">d</font><b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green" title="E52">A1</font></i>: 
(  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cell</a> <font color="Maroon" title="c2">l</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cell</a> <font color="Maroon" title="c5">l'</font>,<font color="Maroon" title="c6">r'</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds <br/> ( <font color="Maroon" title="c6">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c4">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Olive" title="b1">i</font> ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#D11" target="_self" title="CHAIN_1:def.11">Def11</a></i>;<br/><a NAME="E4:73_1"/>
( <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">l'</font> &amp; <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r'</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:73_1"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="chain_1.html#T32" target="_self" title="CHAIN_1:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E5:73_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds <br/> ( <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c4">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Olive" title="b1">i</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:73_1"><i><font color="Green" title="E52">A1</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:73"/><i><font color="Green" title="E52">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font> holds <br/> ( <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is    <a href="chain_1.html#M2" title="CHAIN_1:mode.2">Gap</a> of <font color="Maroon" title="c4">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Olive" title="b1">i</font> )
 ;<br/>

<a NAME="E3:73"/><b>then </b>
 <a href="chain_1.html#K4" title="CHAIN_1:func.4">cell</a> <font color="Maroon" title="c2">l</font>,<font color="Maroon" title="c3">r</font> is   <a href="chain_1.html#NM3" title="CHAIN_1:NM.3">Cell</a> of <font color="Maroon" title="c1">d</font>,<font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="ref" href="chain_1.html#T34" target="_self" title="CHAIN_1:th.34">Th34</a></i>;<br/>
<b>hence </b><a NAME="E4:73"/>
 <a href="chain_1.html#K4" title="CHAIN_1:func.4">cell</a> <font color="Maroon" title="c2">l</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K9" title="CHAIN_1:func.9">infinite-cell</a> <font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E2:73"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="chain_1.html#D11" target="_self" title="CHAIN_1:def.11">Def11</a></i>;<br/>


</div>
