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

<b>let </b><font color="Maroon">d'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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>assume </b><a NAME="E1:83"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:83"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">d'</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">d'</font>,<font color="Maroon">G</font>;<br/>

<b>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>, <font color="Maroon">i0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E4:83"/><i><font color="Green">E52</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; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <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> ) ) )
 <b>by </b><i><a class="ref" href="chain_1.html#T5" target="_self">Th5</a>, </i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> )
 ;<br/><b>hence </b><a NAME="E2:83_1"/>
<font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:83"/>
 ex <font color="Olive">B1</font>, <font color="Olive">B2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B1</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font> &amp; <font color="Olive">B2</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font> &amp; <font color="Olive">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">B2</font> &amp; ( for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font> or <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">B1</font> or <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">B2</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>proof </b></a><div class="add">

<a NAME="E1:83_2"/>
 ex <font color="Olive">l1</font>, <font color="Olive">r1</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/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> &amp; <font color="Olive">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( ( <font color="Olive">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Olive">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r1</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> ) ) ) or 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">r1</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="Olive">l1</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="Olive">l1</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="Olive">r1</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> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">l1i0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:83_2_1"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">l1i0</font>,<span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_1_1"/>
( <font color="Maroon">l1i0</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="Maroon">i0</font> or <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l1i0</font> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_1_1"><b>suppose </b></a><a NAME="E1:83_2_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">l1i0</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="Maroon">i0</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1i0</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/><a NAME="E2:83_2_1_1_1"/><i><font color="Green">E61</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>  ex <font color="Olive">li</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">li</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<a NAME="E1:83_2_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> )
 
;<br/>
<b>hence </b><a NAME="E2:83_2_1_1_1_1"/>
 ex <font color="Olive">li</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">li</font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">l1</font> being   <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> such that </b><br/><a NAME="E4:83_2_1_1_1"/><i><font color="Green">E63</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E4:83_2_1_1_1"><i><font color="Green">E63</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">l1</font> = <font color="Maroon">l1</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">l1</font>
;<br/><b>take </b>
<font color="Maroon">r</font>
;<br/><b>thus </b><a NAME="E6:83_2_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</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="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( ( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r</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> ) ) ) or 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">l1</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">l1</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="txt" href="chain_1.html#E2:83_2_1_1_1"><i><font color="Green">E61</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_1_1_2"><b>suppose </b></a><a NAME="E1:83_2_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l1i0</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">l1</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">r1</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="E3:83_2_1_1_2"/><i><font color="Green">E65</font></i>: 
(  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l1</font>,<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">G</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">r1</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">l1</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">l1</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">r1</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/>;<br/><b>take </b>
<font color="Maroon">l1</font>
;<br/><b>take </b>
<font color="Maroon">r1</font>
;<br/><a NAME="E4:83_2_1_1_2"/>
( <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:83_2_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( ( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r1</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> ) ) ) or 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">r1</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">l1</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">l1</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">r1</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>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">l1</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">r1</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="E3:83_2"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( ( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r1</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> ) ) ) or 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">r1</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">l1</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">l1</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">r1</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> ) ) )
 ;<br/>
<div><i><font color="Green">E121</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E2:83_2_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> )
 ;<br/><b>hence </b><a NAME="E3:83_2_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E5:83_2"/><i><font color="Green">E123</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 <font color="Maroon">l1</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">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or 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">r1</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">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_3_1"/>
( ( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r1</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> ) ) ) or 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">r1</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">l1</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">l1</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">r1</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> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_3_1_1"><b>suppose </b></a><a NAME="E1:83_2_3_1_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r1</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> ) ) )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_3_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_3_1_1_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> implies <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E2:83_2_3_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> )
 ;<br/><b>hence </b><a NAME="E3:83_2_3_1_1_1"/>
<font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:83_2_3_1_1"/>
( 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">l1</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">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or 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">r1</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">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_3_1_2"><b>suppose </b></a><a NAME="E1:83_2_3_1_2"/>
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">r1</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">l1</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">l1</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">r1</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_2_3_1_2"/>
( 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">l1</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">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or 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">r1</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">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B1</font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l1</font>,<font color="Maroon">r1</font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">d</font>,<font color="Maroon">G</font> <b>by </b><i>, </i>;<br/>
<a NAME="E7:83_2"/>
 ex <font color="Olive">l2</font>, <font color="Olive">r2</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/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> &amp; <font color="Olive">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( ( <font color="Olive">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Olive">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Olive">r2</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> ) ) ) or 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">r2</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="Olive">l2</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="Olive">l2</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="Olive">r2</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> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_4"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">r2i0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:83_2_4"/><i><font color="Green">E126</font></i>: 
<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="Maroon">i0</font></span>)</span>,<font color="Maroon">r2i0</font></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="Maroon">i0</font>
 <b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_4_1"/>
( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2i0</font> or <font color="Maroon">r2i0</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="Maroon">i0</font> )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_4_1_1"><b>suppose </b></a><a NAME="E1:83_2_4_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2i0</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>,  <a href="real_1.html#NM1">Real</a>] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2i0</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/><a NAME="E2:83_2_4_1_1"/><i><font color="Green">E128</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>  ex <font color="Olive">ri</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">ri</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/>

<a NAME="E1:83_2_4_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> )
 
;<br/>
<b>hence </b><a NAME="E2:83_2_4_1_1_1"/>
 ex <font color="Olive">ri</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>,<font color="Olive">ri</font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">r2</font> being   <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> such that </b><br/><a NAME="E4:83_2_4_1_1"/><i><font color="Green">E129</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">r2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">r2</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font> <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">l</font>
;<br/><b>take </b>
<font color="Maroon">r2</font>
;<br/><b>thus </b><a NAME="E6:83_2_4_1_1"/>
( <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="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <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">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r2</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> ) ) ) or 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">r2</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">r2</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#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_4_1_2"><b>suppose </b></a><a NAME="E1:83_2_4_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">r2i0</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="Maroon">i0</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">l2</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">r2</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="E3:83_2_4_1_2"/><i><font color="Green">E132</font></i>: 
(  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l2</font>,<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">G</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">r2</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">l2</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">l2</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">r2</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/>;<br/><b>take </b>
<font color="Maroon">l2</font>
;<br/><b>take </b>
<font color="Maroon">r2</font>
;<br/><a NAME="E4:83_2_4_1_2"/>
( <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E5:83_2_4_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( ( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r2</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> ) ) ) or 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">r2</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">l2</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">l2</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">r2</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="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">l2</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">r2</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="E9:83_2"/><i><font color="Green">E133</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; ( ( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r2</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> ) ) ) or 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">r2</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">l2</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">l2</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">r2</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> ) ) )
 ;<br/>
<div><i><font color="Green">E134</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_5"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> implies <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E2:83_2_5"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> )
 ;<br/><b>hence </b><a NAME="E3:83_2_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:83_2"/>
( 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">l2</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">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or 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">r2</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">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_6_1"/>
( ( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r2</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> ) ) ) or 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">r2</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">l2</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">l2</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">r2</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> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_6_1_1"><b>suppose </b></a><a NAME="E1:83_2_6_1_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r2</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> ) ) )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_6_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_6_1_1_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> implies <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/><a NAME="E2:83_2_6_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> )
 ;<br/><b>hence </b><a NAME="E3:83_2_6_1_1_1"/>
<font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="chain_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:83_2_6_1_1"/>
( 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">l2</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">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or 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">r2</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">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_6_1_2"><b>suppose </b></a><a NAME="E1:83_2_6_1_2"/>
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">r2</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">l2</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">l2</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">r2</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_2_6_1_2"/>
( 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">l2</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">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or 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">r2</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">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B2</font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l2</font>,<font color="Maroon">r2</font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">d</font>,<font color="Maroon">G</font> <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">B1</font>
;<br/>

<b>take </b>
<font color="Maroon">B2</font>
;<br/>

<a NAME="E13:83_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_7_1"/>
( ( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r1</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> ) ) ) or 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">r1</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">l1</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">l1</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">r1</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> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_7_1_1"><b>suppose </b></a><a NAME="E1:83_2_7_1_1"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r1</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> ) ) )
 ;<br/><div class="add"><div><i><font color="Green">E139</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_7_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E2:83_2_7_1_1_1"/>
<font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_7_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_7_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E2:83_2_7_1_1_2"/>
( <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="chain_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:83_2_7_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_7_1_2"><b>suppose </b></a><a NAME="E1:83_2_7_1_2"/>
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">r1</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">l1</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">l1</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">r1</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_2_7_1_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E14:83_2"/>
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T5" target="_self">Th5</a></i>;<br/>

<a NAME="E15:83_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_8"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_8_1"/>
( ( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r2</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> ) ) ) or 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">r2</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">l2</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">l2</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">r2</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> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_8_1_1"><b>suppose </b></a><a NAME="E1:83_2_8_1_1"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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>  st <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> holds <br/>( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r2</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> ) ) )
 ;<br/><div class="add"><div><i><font color="Green">E141</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_8_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_8_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E2:83_2_8_1_1_1"/>
<font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_8_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_8_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i><a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a></i>;<br/><b>hence </b><a NAME="E2:83_2_8_1_1_2"/>
( <font color="Maroon">l2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="chain_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:83_2_8_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_8_1_2"><b>suppose </b></a><a NAME="E1:83_2_8_1_2"/>
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">r2</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">l2</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">l2</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">r2</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:83_2_8_1_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E16:83_2"/>
<font color="Maroon">B2</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T5" target="_self">Th5</a></i>;<br/>

<a NAME="E17:83_2"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">l1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:83_2"/>
( 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">l1</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">r1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> or 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">r1</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">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E19:83_2"/>
<font color="Maroon">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="txt" href="chain_1.html#E3:83_2"><i><font color="Green">E66</font></i></a>, </i>;<br/>

<b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E20:83_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="chain_1.html#NM3">Cell</a> of <font color="Maroon">d</font>,<font color="Maroon">G</font> <b>by </b><i><a class="ref" href="chain_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>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="E23:83_2"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">B</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; ( 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  <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> ) &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">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> or 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">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> ) )
 <b>by </b><i/>;<br/>
<a NAME="E24:83_2"/><i><font color="Green">E184</font></i>: 
( <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="Maroon">i0</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="Maroon">i0</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="Maroon">i0</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</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="Maroon">i0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E25:83_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="chain_1.html#T6" target="_self">Th6</a>, <a class="txt" href="chain_1.html#E4:83"><i><font color="Green">E52</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_9"/>
( 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">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> or 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">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> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9_1"><b>suppose </b></a><a NAME="E1:83_2_9_1"/><i><font color="Green">E186</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 <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>
 ;<br/><div class="add"><div><i><font color="Green">E187</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><b>assume </b><a NAME="E1:83_2_9_1_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font>
 ;<br/><a NAME="E2:83_2_9_1_1"/>
<font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">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="Maroon">i</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:83_2_9_1_1"/><b>then </b>
( ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) or ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) or ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, , , <a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, <a class="ref" href="chain_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E4:83_2_9_1_1"/>
( <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E3:83_2_9_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_9_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:83_2_9_1_2"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</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="Maroon">i0</font>
 
<b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_9_1_2_1"/>
( ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> ) or ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> ) )
 </span><b>by </b><i>, , , <a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, , <a class="ref" href="chain_1.html#D2" target="_self">Def2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9_1_2_1_1"><b>suppose </b></a><a NAME="E1:83_2_9_1_2_1_1"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 ;<br/><div class="add"><a NAME="E2:83_2_9_1_2_1_1"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, </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>, <font color="Maroon">l1</font> = <font color="Maroon">l1</font>, <font color="Maroon">r1</font> = <font color="Maroon">r1</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_9_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_9_1_2_1_1_1"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i0</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="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, , </i>;<br/><a NAME="E2:83_2_9_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, <a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E3:83_2_9_1_2_1_1_1"/>
<font color="Maroon">l'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , </i>;<br/><a NAME="E4:83_2_9_1_2_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, <a class="ref" href="chain_1.html#D4" target="_self">Def4</a>, </i>;<br/><b>hence </b><a NAME="E5:83_2_9_1_2_1_1_1"/>
<font color="Maroon">r'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E5:83_2_9_1_2_1_1"/><b>then </b>
( <font color="Maroon">l'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">r'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r1</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E6:83_2_9_1_2_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9_1_2_1_2"><b>suppose </b></a><a NAME="E1:83_2_9_1_2_1_2"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 ;<br/><div class="add"><a NAME="E2:83_2_9_1_2_1_2"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, </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>, <font color="Maroon">l2</font> = <font color="Maroon">l2</font>, <font color="Maroon">r2</font> = <font color="Maroon">r2</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_9_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>;<br/><a NAME="E1:83_2_9_1_2_1_2_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, , </i>;<br/><a NAME="E2:83_2_9_1_2_1_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">r2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, <a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E3:83_2_9_1_2_1_2_1"/>
<font color="Maroon">r'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, , </i>;<br/><a NAME="E4:83_2_9_1_2_1_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">l'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> ) )
 <b>by </b><i>, <a class="ref" href="chain_1.html#D4" target="_self">Def4</a>, </i>;<br/><b>hence </b><a NAME="E5:83_2_9_1_2_1_2_1"/>
<font color="Maroon">l'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E5:83_2_9_1_2_1_2"/><b>then </b>
( <font color="Maroon">l'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">l2</font> &amp; <font color="Maroon">r'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E6:83_2_9_1_2_1_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9_2"><b>suppose </b></a><a NAME="E1:83_2_9_2"/><i><font color="Green">E196</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 <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>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E3:83_2_9_2"/><i><font color="Green">E197</font></i>: 
( ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> ) or ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> ) )
 <b>by </b><i>, , , <a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E4:83_2_9_2"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#D4" target="_self">Def4</a></i>;<br/><b>thus </b><a NAME="E5:83_2_9_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_9_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_9_2_1_1"/>
( ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> ) or ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> ) )
 </span><b>by </b><i><a class="ref" href="chain_1.html#D4" target="_self">Def4</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9_2_1_1_1"><b>suppose </b></a><a NAME="E1:83_2_9_2_1_1_1"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 ;<br/><div class="add"><a NAME="E2:83_2_9_2_1_1_1"/><b>then </b>
<font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:83_2_9_2_1_1_1"/><b>then </b>
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="chain_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:83_2_9_2_1_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, <a class="ref" href="chain_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_9_2_1_1_2"><b>suppose </b></a><a NAME="E1:83_2_9_2_1_1_2"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 ;<br/><div class="add"><a NAME="E2:83_2_9_2_1_1_2"/><b>then </b>
<font color="Maroon">r'</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:83_2_9_2_1_1_2"/><b>then </b>
<font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K9" target="_self">infinite-cell</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="chain_1.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E4:83_2_9_2_1_1_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, <a class="ref" href="chain_1.html#T9" target="_self">Th9</a>, <a class="ref" href="chain_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<b>hence </b><a NAME="E7:83"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K10" target="_self">star</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i/>;<br/>


</div>
