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<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>given </b><font color="Maroon">p1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P1</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P2</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><a NAME="E1:10"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">P1</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">P2</font> <a href="topreal4.html#R1" target="_self">is_S-P_arc_joining</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal4.html#D2" target="_self">TOPREAL4:def 2</a><br/>

<b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P1</font>, <font color="Maroon">P2</font> = <font color="Maroon">P2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E5:10"/>
( <font color="Maroon">P1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">P2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:10"/>
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 <b>by </b><i>, , <a class="ref" href="topreal2.html#T6">TOPREAL2:6</a></i>;<br/>


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