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

<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">k'</font> be    <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="c3">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="c4">l</font>, <font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">l'</font>, <font color="Maroon" title="c7">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="c3">d</font>;<br/><b>let </b><font color="Maroon" title="c8">G</font> be    <a href="chain_1.html#M1" title="CHAIN_1:mode.1">Grating</a> of <font color="Maroon" title="c3">d</font>;<br/>















<b>assume </b><a NAME="E1:63"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">k'</font> &amp; <font color="Maroon" title="c2">k'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">d</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> <font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c8">G</font> &amp;  <a href="chain_1.html#K4" title="CHAIN_1:func.4">cell</a> <font color="Maroon" title="c6">l'</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c2">k'</font>,<font color="Maroon" title="c8">G</font> )
 ;<br/>

<a NAME="E2:63"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E3:63"/><i><font color="Green" title="E49">A3</font></i>: 
 <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="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="c6">l'</font>,<font color="Maroon" title="c7">r'</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">X</font> being   <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="c3">d</font></span>)</span><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green" title="E50">A4</font></i>: 
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</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="c3">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="c9">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="c8">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="c9">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="c8">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#E1:63"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="chain_1.html#E2:63"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="chain_1.html#T34" target="_self" title="CHAIN_1:th.34">Th34</a></i>;<br/>
<a NAME="E6:63"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="chain_1.html#E2:63"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">X</font></span>)</span>
<b>by </b><i><a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
<font color="Maroon" title="c3">d</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="chain_1.html#E5:63"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c10">i0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:63"/><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c10">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c9">X</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E6:63"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">i0</font> = <font color="Maroon" title="c10">i0</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="c3">d</font> <b>by </b><i><a class="txt" href="chain_1.html#E9:63"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:63"/>
not <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X</font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E9:63"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E12:63"/><b>then </b><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</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">i0</font>
 
<b>by </b><i><a class="txt" href="chain_1.html#E5:63"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_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">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> &amp; <font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> ) or ( <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> &amp; <font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> ) or ( <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> &amp; <font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> ) or <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> )
 </span><b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="chain_1.html#E2:63"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="chain_1.html#E3:63"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="chain_1.html#T46" target="_self" title="CHAIN_1:th.46">Th46</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1"><b>suppose </b></a><a NAME="E1:63_2_1"/>
( <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> &amp; <font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_2_1"/>
 ex <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="c3">d</font> st <br/>( ( <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="c6">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="c5">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="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) or ( <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="c7">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="c5">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="c7">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#E12:63"><i><font color="Green" title="E53">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2"><b>suppose </b></a><a NAME="E1:63_2_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">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> &amp; <font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_2_2"/>
 ex <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="c3">d</font> st <br/>( ( <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="c6">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="c5">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="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) or ( <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="c7">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="c5">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="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_3"><b>suppose </b></a><a NAME="E1:63_2_3"/>
( <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> &amp; <font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_2_3"/>
 ex <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="c3">d</font> st <br/>( ( <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="c6">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="c5">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="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) or ( <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="c7">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="c5">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="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_4"><b>suppose </b></a><a NAME="E1:63_2_4"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:63_2_4"/><i><font color="Green" title="E55">A9</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="c3">d</font> holds <br/> ( ( <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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> or <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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">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="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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> or <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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Olive" title="b1">i</font> ) )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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="c3">d</font>,  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>] <b>means </b>( <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $2 &amp; $2 <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> $1 &amp; <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> $2 &amp; $2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> $1 );<br/><a NAME="E3:63_2_4"/><i><font color="Green" title="E56">A10</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="c3">d</font>  ex <font color="Olive" title="b2">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">xi</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">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="c3">d</font>;<br/>

<a NAME="E1:63_2_4_1"/><i><font color="Green" title="E57">A11</font></i>: 
( <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">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="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">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="c8">G</font> <a href="chain_1.html#K3" title="CHAIN_1:func.3">.</a> <font color="Maroon" title="c12">i</font> )
 
<b>by </b><i><a class="txt" href="chain_1.html#E1:63"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="chain_1.html#E2:63"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="chain_1.html#E1:63_2_4"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="chain_1.html#T35" target="_self" title="CHAIN_1:th.35">Th35</a>, <a class="ref" href="chain_1.html#T36" target="_self" title="CHAIN_1:th.36">Th36</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_4_1_1"/>
( ( <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">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="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> ) or <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> or <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:63_2_4_1_1_1"/><i><font color="Green" title="E58">A12</font></i>: 
( <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">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="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
;<br/><b>thus </b><a NAME="E2:63_2_4_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>]
 <b>by </b><i><a class="txt" href="chain_1.html#E5:63"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="chain_1.html#E1:63_2_4_1_1_1"><i><font color="Green" title="E58">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:63_2_4_1_1_2"/><i><font color="Green" title="E58">A13</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 ;<br/><div class="add"><a NAME="E2:63_2_4_1_1_2"/><i><font color="Green" title="E59">A14</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63_2_4_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></i>;<br/><a NAME="E3:63_2_4_1_1_2"/><b>then </b><i><font color="Green" title="E60">A15</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63_2_4_1_1_2"><i><font color="Green" title="E58">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:63_2_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63_2_4"><i><font color="Green" title="E55">A9</font></i></a></i>;<br/><a NAME="E5:63_2_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">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="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E5:63"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="chain_1.html#E3:63_2_4_1_1_2"><i><font color="Green" title="E60">A15</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">xi</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:63_2_4_1_1_2"/><i><font color="Green" title="E61">A16</font></i>: 
( <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">xi</font> &amp; <font color="Maroon" title="c13">xi</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="c12">i</font> )
 <b>by </b><i><a class="ref" href="chain_1.html#T1" target="_self" title="CHAIN_1:th.1">Th1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c13">xi</font>
;<br/><a NAME="E8:63_2_4_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63_2_4_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E9:63_2_4_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c13">xi</font>]
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63_2_4_1_1_2"><i><font color="Green" title="E59">A14</font></i></a>, <a class="txt" href="chain_1.html#E7:63_2_4_1_1_2"><i><font color="Green" title="E61">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_4_1_1_3"><b>suppose </b></a><a NAME="E1:63_2_4_1_1_3"/><i><font color="Green" title="E58">A17</font></i>: 
<font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 ;<br/><div class="add"><a NAME="E2:63_2_4_1_1_3"/>
<font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">l</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63_2_4_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></i>;<br/><a NAME="E3:63_2_4_1_1_3"/><b>then </b><i><font color="Green" title="E59">A18</font></i>: 
<font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">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="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63_2_4_1_1_3"><i><font color="Green" title="E58">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E4:63_2_4_1_1_3"/>
<font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">r</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:63_2_4_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="chain_1.html#T17" target="_self" title="CHAIN_1:th.17">Th17</a></i>;<br/><a NAME="E5:63_2_4_1_1_3"/><b>then </b>
<font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c12">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="c12">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E5:63"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="chain_1.html#E3:63_2_4_1_1_3"><i><font color="Green" title="E59">A18</font></i></a></i>;<br/><b>hence </b><a NAME="E6:63_2_4_1_1_3"/>
 ex <font color="Olive" title="b1">xi</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">i</font>,<font color="Olive" title="b1">xi</font>]
 <b>by </b><i><a class="txt" href="chain_1.html#E2:63_2_4"><i><font color="Green" title="E55">A9</font></i></a>, <a class="txt" href="chain_1.html#E3:63_2_4_1_1_3"><i><font color="Green" title="E59">A18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c12">x</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">d</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E5:63_2_4"/><i><font color="Green" title="E57">A19</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="c3">d</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c12">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E3:63_2_4"><i><font color="Green" title="E56">A10</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</font> as    <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="c3">d</font> <b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self" title="CHAIN_1:def.4">Def4</a></i>;<br/><a NAME="E7:63_2_4"/><i><font color="Green" title="E58">A20</font></i>: 
<font color="Maroon" title="c13">x</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="c4">l</font>,<font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E5:63_2_4"><i><font color="Green" title="E57">A19</font></i></a></i>;<br/><a NAME="E8:63_2_4"/>
( <font color="Maroon" title="c7">r'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">l'</font> <a href="chain_1.html#K2" title="CHAIN_1:func.2">.</a> <font color="Maroon" title="c11">i0</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="c3">d</font>  st <font color="Maroon" title="c7">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="c6">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="c7">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="c13">x</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="c13">x</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="c6">l'</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#E1:63_2_4"><i><font color="Green" title="E54">A8</font></i></a>, <a class="txt" href="chain_1.html#E5:63_2_4"><i><font color="Green" title="E57">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E9:63_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="chain_1.html#E3:63"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="chain_1.html#E7:63_2_4"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="chain_1.html#T26" target="_self" title="CHAIN_1:th.26">Th26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
