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

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:45"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:45"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:45"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:45"/><i><font color="Green">E44</font></i>: 
8 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E6:45"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E7:45"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> holds <br/> <a href="jordan6.html#R1">LE</a> <font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>,<font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E8:45"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">C</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E9:45"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st <font color="Olive">W</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">C</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E10:45"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E11:45"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="jordan7.html#K1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E12:45"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><a href="jordan7.html#K1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E13:45"/><i><font color="Green">E60</font></i>: 
 <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E14:45"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> &amp; not <font color="Olive">i</font>,<font color="Olive">j</font> <a href="gobrd10.html#R1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E15:45"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> holds <br/> <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T21">JORDAN7:21</a></i>;<br/>
<a NAME="E16:45"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E17:45"/><b>then </b>
<font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E18:45"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E19:45"/><b>then </b>
<font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as    <a href="jordan_a.html#M1" target="_self">Segmentation</a> of <font color="Maroon">C</font> <b>by </b><i>, , , , <a class="txt" href="jordan_a.html#E3:45"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan_a.html#E4:45"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan_a.html#E5:45"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan_a.html#E6:45"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan_a.html#E7:45"><i><font color="Green">E46</font></i></a>, , </i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<a NAME="E21:45"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">h</font>,<font color="Olive">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span><b> such that </b><br/><a NAME="E2:45_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">h</font>,<font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E3:45_1"/><i><font color="Green">E64</font></i>: 
 <a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> ) or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_1"/><i><font color="Green">E66</font></i>: 
( 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#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> )
 ;<br/><div class="add"><a NAME="E2:45_1_1_1"/><b>then </b>
 <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">h</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:45_1_1_1"/>
 <a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan_a.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan_a.html#T2" target="_self">Th2</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_2"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> )
 ;<br/><div class="add"><a NAME="E2:45_1_1_2"/><b>then </b>
 <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">h</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:45_1_1_2"/>
 <a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">h</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="jordan_a.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E22:45"/>
 <a href="jordan_a.html#K4" target="_self">diameter</a> <font color="Maroon">h</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/>


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