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<span class="kw">let </span><font color="Maroon" title="c1">P</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> is  <a href="topreal4.html#V1" title="TOPREAL4:attr.1">being_special_polygon</a>  implies <font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</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>, <font color="Maroon" title="c4">P1</font>, <font color="Maroon" title="c5">P2</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><span class="kw"> such that </span><a NAME="E2:9"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
( <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="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</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">P</font> )
 <span class="kw">and </span><br/><a NAME="E3:9"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
( <font color="Maroon" title="c4">P1</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c5">P2</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 <span class="kw">and </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c4">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">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="c2">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:9"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">P2</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="topreal4.html#D2" target="_self" title="TOPREAL4:def.2">TOPREAL4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">P1</font> = <font color="Maroon" title="c4">P1</font>, <font color="Maroon" title="c7">P2</font> = <font color="Maroon" title="c5">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a></span>;<br/>
<a NAME="E7:9"/>
( <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="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp; <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="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="topreal4.html#T3" target="_self" title="TOPREAL4:th.3">Th3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:9"/>
<font color="Maroon" title="c1">P</font> is  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="topreal2.html#T6" title="TOPREAL2:th.6">TOPREAL2:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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