<?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>, <font color="Maroon" title="c4">l'</font>, <font color="Maroon" title="c5">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="c6">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/><b>let </b><font color="Maroon" title="c7">X</font>, <font color="Maroon" title="c8">X'</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span>;<br/>













<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E49">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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cell</a> <font color="Maroon" title="c4">l'</font>,<font color="Maroon" title="c5">r'</font>
 ;<br/>

<b>assume </b><a NAME="E2:64"/><i><font color="Green" title="E50">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="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <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> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <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> &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="c6">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Olive" title="b1">i</font> ) or ( not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Olive" title="b1">i</font> ) )
 ;<br/>

<b>assume </b><a NAME="E3:64"/><i><font color="Green" title="E51">A3</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="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X'</font> &amp; <font color="Maroon" title="c4">l'</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">r'</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="c4">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="c5">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="c6">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Olive" title="b1">i</font> ) or ( not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X'</font> &amp; <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Olive" title="b1">i</font> ) )
 ;<br/>

<a NAME="E4:64"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> &amp; <font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> )
 
<b>by </b><i><a class="ref" href="chain_1.html#T27" target="_self" title="CHAIN_1:th.27">Th27</a></i>;<br/>
<a NAME="E5:64"/><i><font color="Green" title="E53">A5</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  <font color="Maroon" title="c4">l'</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">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E3:64"><i><font color="Green" title="E51">A3</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:64"/>
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">X'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green" title="E54">A6</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; not <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X'</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c9">i</font> as    <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> ;<br/>
<a NAME="E3:64_1"/>
( <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">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="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c10">i</font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:64"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="chain_1.html#E2:64"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="chain_1.html#E3:64"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="chain_1.html#E4:64"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="chain_1.html#E5:64"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="chain_1.html#E1:64_1"><i><font color="Green" title="E54">A6</font></i></a>, <a class="ref" href="chain_1.html#T25" target="_self" title="CHAIN_1:th.25">Th25</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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

<b>set </b><font color="Maroon" title="c9">k</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">X</font>;<br/>
<b>set </b><font color="Maroon" title="c10">k'</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">X'</font>;<br/>
<a NAME="E7:64"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">d</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>;<br/>
<a NAME="E8:64"/><b>then </b><i><font color="Green" title="E54">A7</font></i>: 
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">X</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">d</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">X'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">d</font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T80" title="CARD_1:th.80">CARD_1:80</a></i>;<br/>
<a NAME="E9:64"/><b>then </b><i><font color="Green" title="E55">A8</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#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">cells</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">X</font></span>)</span>,<font color="Maroon" title="c6">G</font> &amp;  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cell</a> <font color="Maroon" title="c4">l'</font>,<font color="Maroon" title="c5">r'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="chain_1.html#K5" title="CHAIN_1:func.5">cells</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">X'</font></span>)</span>,<font color="Maroon" title="c6">G</font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E2:64"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="chain_1.html#E3:64"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="chain_1.html#T34" target="_self" title="CHAIN_1:th.34">Th34</a></i>;<br/>
<b>thus </b><a NAME="E10:64"/>
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>  st ( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> or  not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X'</font> ) holds <br/>( <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font> be    <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>;<br/>

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green" title="E56">A9</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> or  not <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X'</font> )
 ;<br/>

<a NAME="E2:64_2"/>
<font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E3:64"><i><font color="Green" title="E51">A3</font></i></a></i>;<br/>
<a NAME="E3:64_2"/><b>then </b>
( ( <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> ) or ( <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> ) or ( <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> ) )
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:64"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="chain_1.html#E8:64"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="chain_1.html#E9:64"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="chain_1.html#T46" target="_self" title="CHAIN_1:th.46">Th46</a></i>;<br/>
<b>hence </b><a NAME="E4:64_2"/>
( <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E2:64"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="chain_1.html#E3:64"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="chain_1.html#E1:64_2"><i><font color="Green" title="E56">A9</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E11:64"/>
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>  st not <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X'</font> &amp; not <!--HUMANRECFAILED-->( <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) holds <br/>( <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> &amp; <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font> be    <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>;<br/>

<b>assume </b><a NAME="E1:64_3"/>
( not <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X'</font> )
 ;<br/>

<a NAME="E2:64_3"/><b>then </b>
( <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E2:64"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="chain_1.html#E3:64"><i><font color="Green" title="E51">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:64_3"/>
( ( <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> ) or ( <font color="Maroon" title="c2">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c3">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i</font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:64"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="chain_1.html#E8:64"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="chain_1.html#E9:64"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="chain_1.html#T46" target="_self" title="CHAIN_1:th.46">Th46</a></i>;<br/>


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


</div>
