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<b>let </b><font color="Maroon" title="c1">f</font> be  non <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon" title="c2">g</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">i1</font>, <font color="Maroon" title="c4">i2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E54">A1</font></i>: 
( <font color="Maroon" title="c2">g</font> <a href="jordan4.html#R3" title="JORDAN4:pred.3">is_a_part_of</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c3">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">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" title="c2">g</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font> or <font color="Maroon" title="c2">g</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E1:53"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="jordan4.html#D4" target="_self" title="JORDAN4:def.4">Def4</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" title="c2">g</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_1"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:53"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="jordan4.html#T58" target="_self" title="JORDAN4:th.58">Th58</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" title="c2">g</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:53_1_1_2"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:53"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="jordan4.html#T59" target="_self" title="JORDAN4:th.59">Th59</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" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c4">i2</font>
 ;<br/>


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