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<b>let </b><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">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>let </b><font color="Maroon" title="c3">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="c4">g1</font>, <font color="Maroon" title="c5">g2</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>assume </b><a NAME="E1:60"/><i><font color="Green" title="E59">A1</font></i>: 
( <font color="Maroon" title="c1">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="jordan4.html#R3" title="JORDAN4:pred.3">is_a_part_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R3" title="JORDAN4:pred.3">is_a_part_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c4">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1"/>
( <font color="Maroon" title="c4">g1</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> or <font color="Maroon" title="c4">g1</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E1:60"><i><font color="Green" title="E59">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="60_1_1"><b>suppose </b></a><a NAME="E1:60_1_1"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_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="E1:60_1_1_1_1"/>
( <font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> or <font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E1:60"><i><font color="Green" title="E59">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="60_1_1_1_1_1"><b>case </b></a><a NAME="E1:60_1_1_1_1_1"/>
<font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_1_1_1_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:60_1_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="jordan4.html#T64" target="_self" title="JORDAN4:th.64">Th64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_1_1_1_2"><b>case </b></a><a NAME="E1:60_1_1_1_1_2"/>
<font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan4.html#E1:60"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:60_1_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="jordan4.html#T66" target="_self" title="JORDAN4:th.66">Th66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:60_1_1"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2"><b>suppose </b></a><a NAME="E1:60_1_2"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:60_1_2_1_1"/>
( <font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> or <font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> )
 </span><b>by </b><i><a class="txt" href="jordan4.html#E1:60"><i><font color="Green" title="E59">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="60_1_2_1_1_1"><b>case </b></a><a NAME="E1:60_1_2_1_1_1"/>
<font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R1" title="JORDAN4:pred.1">is_a_part&gt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan4.html#E1:60"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="jordan4.html#E1:60_1_2"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="jordan4.html#T66" target="_self" title="JORDAN4:th.66">Th66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="60_1_2_1_1_2"><b>case </b></a><a NAME="E1:60_1_2_1_1_2"/>
<font color="Maroon" title="c5">g2</font> <a href="jordan4.html#R2" title="JORDAN4:pred.2">is_a_part&lt;_of</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:60_1_2_1_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan4.html#E1:60_1_2"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="jordan4.html#T65" target="_self" title="JORDAN4:th.65">Th65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:60_1_2"/>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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