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<div class="add">

<b>let </b><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font> be   <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>let </b><font color="Maroon" title="c3">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <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/>



<b>assume </b><a NAME="E1:90"/><i><font color="Green" title="E79">A1</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a>  &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">P</font> )
 ;<br/>

<a NAME="E2:90"/><b>then </b><i><font color="Green" title="E80">A2</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E3:90"/><i><font color="Green" title="E81">A3</font></i>: 
not <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="jordan7.html#T3" title="JORDAN7:th.3">JORDAN7:3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:90_1"/>
( ( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> ) or ( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> ) or ( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> ) or ( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E2:90"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1"><b>suppose </b></a><a NAME="E1:90_1_1"/><i><font color="Green" title="E82">A4</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><div class="add"><a NAME="E2:90_1_1"/><i><font color="Green" title="E83">A5</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/><b>set </b><font color="Maroon" title="c4">q1</font> =  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>;<br/><b>set </b><font color="Maroon" title="c5">q2</font> =  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>;<br/><b>set </b><font color="Maroon" title="c6">Q</font> =  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_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="E1:90_1_1_1_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_1"><b>case </b></a><a NAME="E1:90_1_1_1_1_1"/><i><font color="Green" title="E84">A6</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_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="E1:90_1_1_1_1_1_1_1"/>
( (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font> ) or (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:90_1_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="txt" href="jgraph_6.html#E2:90_1_1"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="jgraph_6.html#E1:90_1_1_1_1_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="jordan5c.html#T14" title="JORDAN5C:th.14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:90_1_1_1_1_1_1_1_1"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:90_1_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:90_1_1_1_1_1_1_1_2"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_1_1_1_1_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90_1_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:90_1_1_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1_1_1_2"><b>case </b></a><a NAME="E1:90_1_1_1_1_2"/>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_1_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="jordan6.html#T71" title="JORDAN6:th.71">JORDAN6:71</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:90_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2"><b>suppose </b></a><a NAME="E1:90_1_2"/><i><font color="Green" title="E82">A7</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_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="E1:90_1_2_1_1"/>
( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_1_1"><b>case </b></a><a NAME="E1:90_1_2_1_1_1"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_2_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="jordan7.html#T3" title="JORDAN7:th.3">JORDAN7:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2_1_1_2"><b>case </b></a><a NAME="E1:90_1_2_1_1_2"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:90_1_2"><i><font color="Green" title="E82">A7</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:90_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_3"><b>suppose </b></a><a NAME="E1:90_1_3"/>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4"><b>suppose </b></a><a NAME="E1:90_1_4"/><i><font color="Green" title="E82">A8</font></i>: 
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><div class="add"><a NAME="E2:90_1_4"/><i><font color="Green" title="E83">A9</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="jordan6.html#T65" title="JORDAN6:th.65">JORDAN6:65</a></i>;<br/><b>set </b><font color="Maroon" title="c4">q2</font> =  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>;<br/><b>set </b><font color="Maroon" title="c5">q1</font> =  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>;<br/><b>set </b><font color="Maroon" title="c6">Q</font> =  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:90_1_4_1_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_1_1_1"><b>case </b></a><a NAME="E1:90_1_4_1_1_1"/><i><font color="Green" title="E84">A10</font></i>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_4_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_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="E1:90_1_4_1_1_1_1_1"/>
( (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> ) or (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> ) )
 </span><b>by </b><i><a class="txt" href="jgraph_6.html#E1:90_1_4"><i><font color="Green" title="E82">A8</font></i></a>, <a class="txt" href="jgraph_6.html#E2:90_1_4"><i><font color="Green" title="E83">A9</font></i></a>, <a class="txt" href="jgraph_6.html#E1:90_1_4_1_1_1"><i><font color="Green" title="E84">A10</font></i></a>, <a class="ref" href="jordan5c.html#T14" title="JORDAN5C:th.14">JORDAN5C:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:90_1_4_1_1_1_1_1_1"/><i><font color="Green" title="E85">A11</font></i>: 
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_4_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_1_1_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="E1:90_1_4_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:90_1_4_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_4_1_1_1_1_1_1_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="jordan7.html#T3" title="JORDAN7:th.3">JORDAN7:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:90_1_4_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_4_1_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="jgraph_6.html#E1:90_1_4"><i><font color="Green" title="E82">A8</font></i></a>, <a class="txt" href="jgraph_6.html#E1:90_1_4_1_1_1_1_1_1"><i><font color="Green" title="E85">A11</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:90_1_4_1_1_1_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:90_1_4_1_1_1_1_1_2"/>
(  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> &amp; not  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c3">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c3">P</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_4_1_1_1_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E3:90"><i><font color="Green" title="E81">A3</font></i></a>, <a class="txt" href="jgraph_6.html#E1:90_1_4"><i><font color="Green" title="E82">A8</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:90_1_4_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_4_1_1_2"><b>case </b></a><a NAME="E1:90_1_4_1_1_2"/>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_4_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E1:90"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="jordan6.html#T71" title="JORDAN6:th.71">JORDAN6:71</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:90_1_4"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">P</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
