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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#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E50">A1</font></i>: 
( <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> &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="52_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1"/>
( <font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">i2</font> or <font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><b>case </b></a><a NAME="E1:52_1_1_1"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">i2</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_1"/>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <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="c4">i2</font>,<font color="Maroon" title="c3">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:52"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="jordan4.html#T42" target="_self" title="JORDAN4:th.42">Th42</a></i>;<br/><a NAME="E3:52_1_1_1"/><b>then </b><i><font color="Green" title="E52">A3</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c2">g</font></span>)</span> <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="c4">i2</font>,<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>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:52_1_1_1"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="jordan4.html#T56" target="_self" title="JORDAN4:th.56">Th56</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">p2</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>, <font color="Maroon" title="c6">p1</font> = <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> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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>reconsider </b><font color="Maroon" title="c7">P</font> =  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:52_1_1_1"/>
<font color="Maroon" title="c7">P</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c6">p1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E3:52_1_1_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><b>hence </b><a NAME="E7:52_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="ref" href="sppol_2.html#T53" title="SPPOL_2:th.53">SPPOL_2:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2"><b>case </b></a><a NAME="E1:52_1_1_2"/>
<font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1_2"/><b>then </b><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:52"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:52_1_1_2"/>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <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="c4">i2</font>,<font color="Maroon" title="c3">i1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:52"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="jordan4.html#T42" target="_self" title="JORDAN4:th.42">Th42</a></i>;<br/><a NAME="E4:52_1_1_2"/><b>then </b><i><font color="Green" title="E52">A5</font></i>: 
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c2">g</font></span>)</span> <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="c4">i2</font>,<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>
 <b>by </b><i><a class="txt" href="jordan4.html#E2:52_1_1_2"><i><font color="Green" title="E51">A4</font></i></a>, <a class="txt" href="jordan4.html#E61"><i><font color="Green" title="E47">Lm1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">p2</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>, <font color="Maroon" title="c6">p1</font> = <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> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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>reconsider </b><font color="Maroon" title="c7">P</font> =  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c2">g</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E7:52_1_1_2"/>
<font color="Maroon" title="c7">P</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c6">p1</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E4:52_1_1_2"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></i>;<br/><b>hence </b><a NAME="E8:52_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="ref" href="sppol_2.html#T53" title="SPPOL_2:th.53">SPPOL_2:53</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:52"/>
 <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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