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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="jordan_a.html#M1" target="_self">Segmentation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:47"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="gobrd10.html#R1">are_adjacent1</a>  ) </span>}</span> 
 <b>and </b><br/><a NAME="E3:47"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:47"/><i><font color="Green">E39</font></i>: 
 <a href="jordan_a.html#K5" target="_self">S-Gap</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan_a.html#D7" target="_self">Def7</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E5:47"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan_a.html#E1:47_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:47_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan1k.html#K4">dist_min</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:47_1_1_1"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E5:47_1_1_1"/><i><font color="Green">E43</font></i>: 
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="gobrd10.html#R1">are_adjacent1</a> 
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:47"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E6:47_1_1_1"/>
 <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:47_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan_a.html#E5:47_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan_a.html#T25" target="_self">Th25</a></i>;<br/><b>hence </b><a NAME="E7:47_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:47_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan_a.html#E4:47_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:47_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan1k.html#K4">dist_min</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:47_1_1_2"/><i><font color="Green">E42</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:47_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:47"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E6:47_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E5:47_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E7:47_1_1_2"/>
 <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:47_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan_a.html#E5:47_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan_a.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E8:47_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:47_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan_a.html#E4:47_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan_a.html#E6:47_1_1_2"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E6:47"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:47_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan1k.html#K4">dist_min</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:47_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan1k.html#K4">dist_min</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:47_2"/><i><font color="Green">E41</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:47_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E6:47_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:47_2"/><i><font color="Green">E44</font></i>: 
 <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 ;<br/>
<a NAME="E8:47_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E5:47_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan_a.html#E6:47_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_2_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan_a.html#E6:47_2"><i><font color="Green">E43</font></i></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="47_2_1_1"><b>suppose </b></a><a NAME="E1:47_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_2_1_1"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="gobrd10.html#R1">are_adjacent1</a> 
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:47_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan_a.html#E5:47_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan_a.html#E7:47_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan_a.html#E1:47_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan_a.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E3:47_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:47"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan_a.html#E3:47_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan_a.html#E4:47_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan_a.html#E5:47_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan_a.html#E1:47_2_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><b>hence </b><a NAME="E4:47_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_2_1_2"><b>suppose </b></a><a NAME="E1:47_2_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:47_2_1_2"/><b>then </b>
1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E7:47_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan_a.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E3:47_2_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:47_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:47_2_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan_a.html#E8:47_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E5:47_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan_a.html#E7:47_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan_a.html#E1:47_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan_a.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E6:47_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:47_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:47_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:47"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan_a.html#E3:47_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan_a.html#E1:47_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan_a.html#E3:47_2_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><b>hence </b><a NAME="E8:47_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E7:47"/>
 <a href="jordan_a.html#K5" target="_self">S-Gap</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K3">min</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E4:47"><i><font color="Green">E39</font></i></a>, <a class="ref" href="jordan_a.html#T1" target="_self">Th1</a></i>;<br/>


</div>
