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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</font> being  <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>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; not <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">C</font></span><br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font> &amp; not <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E4:36"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 <b>and </b><br/><a NAME="E5:36"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font>
 <b>and </b><br/><a NAME="E6:36"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 <b>and </b><br/><a NAME="E7:36"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p1</font>
 <b>and </b><br/><a NAME="E8:36"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font> )</span><br/>

<b>assume </b><a NAME="E9:36"/><i><font color="Green" title="E29">A9</font></i>: 
for <font color="Olive" title="b1">A</font> being  <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>  st <font color="Olive" title="b1">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan18.html#D3" target="_self" title="JORDAN18:def.3">JORDAN18:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">P1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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><b> such that </b><br/><a NAME="E11:36"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c6">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>and </b><br/><a NAME="E12:36"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c7">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>and </b><br/><a NAME="E13:36"/><i><font color="Green" title="E32">A12</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P1</font>
 <b>and </b><br/><a NAME="E14:36"/><i><font color="Green" title="E33">A13</font></i>: 
<font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan18.html#E1:36"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="jordan18.html#E2:36"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="jordan18.html#E4:36"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="topreal2.html#T5" title="TOPREAL2:th.5">TOPREAL2:5</a></i>;<br/>
<a NAME="E15:36"/><i><font color="Green" title="E34">A14</font></i>: 
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan18.html#E13:36"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E16:36"/><i><font color="Green" title="E35">A15</font></i>: 
<font color="Maroon" title="c7">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan18.html#E13:36"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E17:36"/><i><font color="Green" title="E36">A16</font></i>: 
<font color="Maroon" title="c6">P</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan18.html#E9:36"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="jordan18.html#E11:36"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="jordan18.html#E13:36"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1"/>
( ( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; not <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) or ( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan18.html#E17:36"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T57" title="ZFMISC_1:th.57">ZFMISC_1:57</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:36_1_1"/><i><font color="Green" title="E37">A17</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <b>and </b><br/><a NAME="E2:36_1_1"/><i><font color="Green" title="E38">A18</font></i>: 
not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c8">A</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> &amp; <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c8">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><a NAME="E1:36_1_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E11:36"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="jordan18.html#E1:36_1_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E2:36_1_1_1"/>
<font color="Maroon" title="c8">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E5:36"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="jordan18.html#E11:36"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="jordan16.html#T36" title="JORDAN16:th.36">JORDAN16:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c8">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><a NAME="E3:36_1_1_1"/><i><font color="Green" title="E39">A19</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1"/>
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E15:36"><i><font color="Green" title="E34">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div><i><font color="Green" title="E40">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_1_1_1"/><i><font color="Green" title="E41">A21</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:36_1_1_1_1"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><a NAME="E3:36_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <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> st <br/>( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan18.html#E1:36_1_1_1_1"><i><font color="Green" title="E41">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan18.html#E6:36"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="jordan18.html#E11:36"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="jordan6.html#T70" title="JORDAN6:th.70">JORDAN6:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E6:36_1_1_1"/>
not <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E2:36_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="jordan18.html#E3:36_1_1_1"><i><font color="Green" title="E39">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E7:36_1_1_1"/>
<font color="Maroon" title="c8">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan18.html#E5:36_1_1_1"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T57" title="ZFMISC_1:th.57">ZFMISC_1:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:36_1_1"/>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="jordan18.html#D3" target="_self" title="JORDAN18:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:36_1_2"/><i><font color="Green" title="E37">A22</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <b>and </b><br/><a NAME="E2:36_1_2"/><i><font color="Green" title="E38">A23</font></i>: 
not <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c8">A</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> &amp; <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c8">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><a NAME="E1:36_1_2_1"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E3:36"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="jordan18.html#E13:36"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="jordan18.html#E2:36_1_2"><i><font color="Green" title="E38">A23</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E2:36_1_2_1"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E12:36"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:36_1_2_1"/>
<font color="Maroon" title="c8">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E5:36"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="jordan18.html#E12:36"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="jordan16.html#T36" title="JORDAN16:th.36">JORDAN16:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c8">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><a NAME="E4:36_1_2_1"/><i><font color="Green" title="E39">A24</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P1</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></i>;<br/><b>hence </b><a NAME="E5:36_1_2_1"/>
<font color="Maroon" title="c8">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E16:36"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div><i><font color="Green" title="E40">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_2_1_1"/><i><font color="Green" title="E41">A26</font></i>: 
<font color="Maroon" title="c3">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:36_1_2_1_1"/>
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><a NAME="E3:36_1_2_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <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> st <br/>( <font color="Maroon" title="c3">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c2">p1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c7">P1</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <b>by </b><i><a class="txt" href="jordan18.html#E1:36_1_2_1_1"><i><font color="Green" title="E41">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan18.html#E6:36"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="jordan18.html#E12:36"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="jordan6.html#T70" title="JORDAN6:th.70">JORDAN6:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_1_2_1_2"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:36_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan18.html#E14:36"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="jordan18.html#E1:36_1_2"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="jordan18.html#E4:36_1_2_1"><i><font color="Green" title="E39">A24</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:36_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan18.html#E7:36"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="jordan18.html#E8:36"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E8:36_1_2_1"/>
<font color="Maroon" title="c8">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan18.html#E6:36_1_2_1"><i><font color="Green" title="E40">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T57" title="ZFMISC_1:th.57">ZFMISC_1:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:36_1_2"/>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="jordan18.html#D3" target="_self" title="JORDAN18:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:36_1_3"/><i><font color="Green" title="E37">A27</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <b>and </b><br/><a NAME="E2:36_1_3"/><i><font color="Green" title="E38">A28</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font></span><br/><div class="add"><a NAME="E3:36_1_3"/>
<font color="Maroon" title="c7">P1</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan18.html#E9:36"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="jordan18.html#E12:36"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="jordan18.html#E13:36"><i><font color="Green" title="E32">A12</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:36_1_3"/><b>then </b>
( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> or <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P1</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T57" title="ZFMISC_1:th.57">ZFMISC_1:57</a></i>;<br/><a NAME="E5:36_1_3"/><b>then </b>
( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="jordan18.html#E14:36"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="jordan18.html#E1:36_1_3"><i><font color="Green" title="E37">A27</font></i></a>, <a class="txt" href="jordan18.html#E2:36_1_3"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E6:36_1_3"/>
<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">q1</font> <a href="jordan18.html#NR2" title="JORDAN18:NR.2">are_neighbours_wrt</a> <font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan18.html#E5:36"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="jordan18.html#E6:36"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="jordan18.html#E7:36"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="jordan18.html#E8:36"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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