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

<b>let </b><font color="Maroon">f</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">g</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">g</font> <a href="jordan4.html#R3" target="_self">is_a_part_of</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1"/>
( <font color="Maroon">g</font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> or <font color="Maroon">g</font> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 </span><b>by </b><i>, <a class="ref" href="jordan4.html#T2" target="_self">Th2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>case </b></a><a NAME="E1:53_1_1_1"/>
<font color="Maroon">g</font> <a href="jordan4.html#R1" target="_self">is_a_part&gt;_of</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="jordan4.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><b>case </b></a><a NAME="E1:53_1_1_2"/>
<font color="Maroon">g</font> <a href="jordan4.html#R2" target="_self">is_a_part&lt;_of</a> <font color="Maroon">f</font>,<font color="Maroon">i1</font>,<font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_2"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="jordan4.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:53"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="topreal4.html#R1">is_S-P_arc_joining</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font>
 ;<br/>


</div>
