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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green">E55</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><b>case </b></a><a NAME="E1:55_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:55_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:55_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</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>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E4:55_1_1_1"/><i><font color="Green">E59</font></i>: 
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan4.html#E1:55_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan4.html#E3:55_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan4.html#T43" target="_self">Th43</a></i>;<br/><a NAME="E5:55_1_1_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan4.html#E2:55_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan4.html#E3:55_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan4.html#T46" target="_self">Th46</a></i>;<br/><a NAME="E6:55_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
(  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E4:55_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E7:55_1_1_1"/><i><font color="Green">E61</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan4.html#E2:55_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan4.html#E3:55_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan4.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E8:55_1_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:55_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan4.html#E4:55_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="jordan4.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E9:55_1_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan4.html#E6:55_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan4.html#T60" target="_self">Th60</a></i>;<br/><a NAME="E10:55_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:55_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E1:55_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan4.html#D4" target="_self">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:55_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan4.html#T37" target="_self">Th37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:55_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan4.html#E2:55_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan4.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:55_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="jordan4.html#E6:55_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan4.html#E7:55_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan4.html#E8:55_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan4.html#E9:55_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="55_1_1_2"><b>case </b></a><a NAME="E1:55_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:55_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<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>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/><a NAME="E3:55_1_1_2"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan4.html#E1:55_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan4.html#E2:55_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan4.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E4:55_1_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E5:55_1_1_2"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan4.html#E1:55_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan4.html#E2:55_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan4.html#T43" target="_self">Th43</a></i>;<br/><a NAME="E6:55_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="jordan4.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E7:55_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan4.html#E1:55_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan4.html#E2:55_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan4.html#T46" target="_self">Th46</a></i>;<br/><a NAME="E8:55_1_1_2"/><b>then </b>
 <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="jordan4.html#T42" target="_self">Th42</a></i>;<br/><a NAME="E9:55_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
(  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E6:55_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan4.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E10:55_1_1_2"/><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan4.html#E3:55_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan4.html#E4:55_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E11:55_1_1_2"/><i><font color="Green">E63</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan4.html#E6:55_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan4.html#T59" target="_self">Th59</a></i>;<br/><a NAME="E12:55_1_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan4.html#E9:55_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="jordan4.html#T60" target="_self">Th60</a></i>;<br/><a NAME="E13:55_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:55_1_1_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E1:55_1_1_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan4.html#D4" target="_self">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:55_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:55_1_1_2_1_1_1_1"/><b>then </b>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>8</sub></font> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="jordan4.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E3:55_1_1_2_1_1_1_1"/><b>then </b>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan4.html#T40" target="_self">Th40</a></i>;<br/><b>hence </b><a NAME="E4:55_1_1_2_1_1_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_6.html#T29">FINSEQ_6:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:55_1_1_2_1_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:55_1_1_2_1_1_1_2"/><b>then </b>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>8</sub></font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="jordan4.html#T42" target="_self">Th42</a></i>;<br/><a NAME="E3:55_1_1_2_1_1_1_2"/><b>then </b>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:55_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan4.html#T37" target="_self">Th37</a></i>;<br/><b>hence </b><a NAME="E4:55_1_1_2_1_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_6.html#T29">FINSEQ_6:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55_1_1_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K4">Rev</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,1</span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E14:55_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="jordan4.html#E9:55_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan4.html#E10:55_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan4.html#E11:55_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan4.html#E12:55_1_1_2"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>2</sub></font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> holds <br/>(  not <font color="Olive">b<sub>3</sub></font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 ;<br/>


</div>
