<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be   <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>let </b><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">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>assume </b><b>that </b><br/><a NAME="E1:14"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c2">A1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <b>and </b><br/><a NAME="E2:14"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c3">A2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <b>and </b><br/><a NAME="E3:14"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c2">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E4:14"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c3">A2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E5:14"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c2">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">A2</font>
 ;<br/>

<a NAME="E6:14"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p2</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="jordan6.html#T49" title="JORDAN6:th.49">JORDAN6:49</a></i>;<br/>
<a NAME="E7:14"/>
( <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> &amp; <font color="Maroon" title="c5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A1</font> )
 
<b>by </b><i><a class="txt" href="jordan16.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">P1</font>, <font color="Maroon" title="c7">P2</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="E9:14"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c6">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <b>and </b><br/><a NAME="E10:14"/><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c7">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <b>and </b><br/><a NAME="E11:14"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">P2</font>
 <b>and </b><br/><a NAME="E12:14"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c6">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">P2</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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan16.html#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="jordan16.html#E6:14"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="topreal2.html#T5" title="TOPREAL2:th.5">TOPREAL2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">P1</font> = <font color="Maroon" title="c6">P1</font>, <font color="Maroon" title="c9">P2</font> = <font color="Maroon" title="c7">P2</font> as  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> ;<br/>
<a NAME="E14:14"/>
( <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P1</font> or <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P2</font> )
 
<b>by </b><i><a class="txt" href="jordan16.html#E1:14"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="jordan16.html#E3:14"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="jordan16.html#E9:14"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="jordan16.html#E10:14"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="jordan16.html#E11:14"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="jordan16.html#E12:14"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="jordan16.html#T14" target="_self" title="JORDAN16:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E15:14"/>
( <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c2">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">A2</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="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="jordan16.html#E2:14"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="jordan16.html#E4:14"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="jordan16.html#E5:14"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="jordan16.html#E9:14"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="jordan16.html#E10:14"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="jordan16.html#E11:14"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="jordan16.html#E12:14"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="jordan16.html#T14" target="_self" title="JORDAN16:th.14">Th14</a></i>;<br/>


</div>
