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

<b>let </b><font color="Maroon">d</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:84"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">G</font> be    <a href="chain_1.html#M1" target="_self">Grating</a> of <font color="Maroon">d</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="chain_1.html#NM3">Cell</a> of <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<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="E3:84"/><i><font color="Green">E51</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; ( <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> or ( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r</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">l</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">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; ( 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">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) ) )
 <b>by </b><i/>;<br/>
<a NAME="E4:84"/>
 ex <font color="Olive">A1</font>, <font color="Olive">A2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A1</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">B</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">A2</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">B</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Olive">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">A2</font> &amp; ( for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">B</font></span><a href="domain_1.html#K6">}</a></span> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">A1</font> or <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<a NAME="E1:84_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 <br/> ( <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">A1</font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">l</font>,<font color="Maroon">l</font>, <font color="Maroon">A2</font> =  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">r</font>,<font color="Maroon">r</font> as   <a href="chain_1.html#NM3">Cell</a> of 0,<font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">A1</font>
;<br/>

<b>take </b>
<font color="Maroon">A2</font>
;<br/>

<a NAME="E3:84_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:84_1"/>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:84_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">l</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E6:84_1"/>
( <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">B</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">B</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i>, <a class="txt" href="chain_1.html#E12:84_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="chain_1.html#T4" target="_self">Th4</a></i>;<br/>

<b>thus </b><a NAME="E7:84_1"/>
<font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A2</font>
 <b>by </b><i>, <a class="txt" href="chain_1.html#E12:84_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>

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

<b>assume </b><a NAME="E8:84_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="chain_1.html#K11" target="_self">del</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">B</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="chain_1.html#NM3">Cell</a> of 0,<font color="Maroon">G</font> ;<br/>
<a NAME="E10:84_1"/><i><font color="Green">E54</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="txt" href="chain_1.html#E13:84_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="chain_1.html#T4" target="_self">Th4</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">d</font><b> such that </b><br/><a NAME="E12:84_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K4" target="_self">cell</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font> holds  <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="chain_1.html#K3" target="_self">.</a> <font color="Olive">i</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E13:84_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_1_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">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> or ( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r</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">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1"><b>suppose </b></a><a NAME="E1:84_1_1_1"/><i><font color="Green">E63</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>
 ;<br/><div class="add"><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_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="E1:84_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="E2:84_1_1_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="txt" href="chain_1.html#E4:84_1_1_1"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:84_1_1_1"/>
<font color="Maroon">x</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>
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E4:84_1_1_1"/><b>then </b><i><font color="Green">E65</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">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">x</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; (  not <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">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> or  not <font color="Maroon">x</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="txt" href="chain_1.html#E2:84_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E5:84_1_1_1"><i><font color="Green">E66</font></i></a>, , </i>;<br/><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_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/><b>assume </b><a NAME="E1:84_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i0</font>
 ;<br/><a NAME="E2:84_1_1_1_2"/><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">r</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">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x</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="txt" href="chain_1.html#E2:84_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="chain_1.html#E5:84_1_1_1"><i><font color="Green">E66</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:84_1_1_1_2"/>
( <font color="Maroon">x</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">x</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="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><b>thus </b><a NAME="E6:84_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_1_1_1_3_1"/>
( <font color="Maroon">x</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> or <font color="Maroon">x</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> )
 </span><b>by </b><i><a class="ref" href="chain_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:84_1_1_1_3_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</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"><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">l</font> = <font color="Maroon">l</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="84_1_1_1_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="E1:84_1_1_1_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="E2:84_1_1_1_3_1_1_1"/>
<font color="Maroon">x</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="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/></div><b>end;</b></div><a NAME="E4:84_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E5:84_1_1_1_3_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_1_1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:84_1_1_1_3_1_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</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"><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_1_3_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="E1:84_1_1_1_3_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> )
 ;<br/><b>hence </b><a NAME="E2:84_1_1_1_3_1_2_1"/>
<font color="Maroon">x</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="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/></div><b>end;</b></div><a NAME="E4:84_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E5:84_1_1_1_3_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="txt" href="chain_1.html#E1:84_1_1_1"><i><font color="Green">E63</font></i></a></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="84_1_1_2"><b>suppose </b></a><a NAME="E1:84_1_1_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">r</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">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 ;<br/><div class="add"><a NAME="E2:84_1_1_2"/><i><font color="Green">E124</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="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_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="E1:84_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i0</font> &amp; <font color="Maroon">i0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:84_1_1_2_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i0</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E3:84_1_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font>
 ;<br/>


</div><b>end;</b></div><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="E4:84_1_1_2"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">r</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</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">i1</font> &amp; ( <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</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">i1</font> or <font color="Maroon">l</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i1</font> ) )
 <b>by </b><i>, , , , </i>;<br/><a NAME="E5:84_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font>
 <b>by </b><i/>;<br/><a NAME="E6:84_1_1_2"/>
<font color="Maroon">x</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>
 <b>by </b><i/>;<br/><a NAME="E7:84_1_1_2"/><b>then </b><i><font color="Green">E127</font></i>: 
( not <font color="Maroon">x</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; not <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">x</font> <a href="chain_1.html#K2" target="_self">.</a> <font color="Maroon">i0</font> )
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E8:84_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_1_1_2_2_1"/>
( <font color="Maroon">x</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">x</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#T6" target="_self">Th6</a>, , <a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:84_1_1_2_2_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</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"><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">d</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1_2_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="E1:84_1_1_2_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:84_1_1_2_2_1_1_1"/>
<font color="Maroon">x</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="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><a NAME="E4:84_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E5:84_1_1_2_2_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:84_1_1_2_2_1_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">x</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"><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">l</font> = <font color="Maroon">l</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="84_1_1_2_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="E1:84_1_1_2_2_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i0</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:84_1_1_2_2_1_2_1"/>
<font color="Maroon">x</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="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="chain_1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><a NAME="E4:84_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>hence </b><a NAME="E5:84_1_1_2_2_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> or <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</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><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:84"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" target="_self">del</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">B</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i/>;<br/>


</div>
