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<b>let </b><font color="Maroon" title="c1">A</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="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>;<br/>









<a NAME="E1:8"/><i><font color="Green" title="E6">A1</font></i>: 
 <a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</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="Olive" title="b1">q</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">A</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#D3" title="JORDAN6:def.3">JORDAN6:def 3</a></i>;<br/>
<a NAME="E2:8"/><i><font color="Green" title="E7">A2</font></i>: 
 <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</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="c4">q1</font>,<font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">A</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#D4" title="JORDAN6:def.4">JORDAN6:def 4</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green" title="E8">A3</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan6.html#D5" title="JORDAN6:def.5">JORDAN6:def 5</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E4:8"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c1">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>
 <b>and </b><br/><a NAME="E5:8"/><i><font color="Green" title="E10">A5</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>
 ;<br/>

<a NAME="E6:8"/><i><font color="Green" title="E11">A6</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">A</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E5:8"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E7:8"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E5:8"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E8:8"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="jordan16.html#E5:8"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/>
<a NAME="E9:8"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E4:8"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="jordan16.html#E6:8"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="jordan16.html#T8" target="_self" title="JORDAN16:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E10:8"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<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>
 <b>by </b><i><a class="txt" href="jordan16.html#E3:8"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="jordan16.html#E8:8"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>

<a NAME="E11:8"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K4" title="JORDAN6:func.4">R_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">q1</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E2:8"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan16.html#E5:8"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/>
<a NAME="E12:8"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K3" title="JORDAN6:func.3">L_Segment</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c5">q2</font>
 
<b>by </b><i><a class="txt" href="jordan16.html#E4:8"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="jordan16.html#E7:8"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="jordan16.html#T7" target="_self" title="JORDAN16:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E13:8"/>
<font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">A</font>,<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>
 <b>by </b><i><a class="txt" href="jordan16.html#E3:8"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="jordan16.html#E11:8"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


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