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<b>let </b><font color="Maroon">D</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">p</font> be   <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>;<br/>



<b>assume </b><a NAME="E1:80"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>consider </b><font color="Maroon">q</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><b> such that </b><br/><a NAME="E3:80"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">P1</font> being  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>, <font color="Maroon">P2</font> being  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> such that </b><br/><a NAME="E5:80"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">P1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E6:80"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E7:80"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P2</font>
 <b>and </b><br/><a NAME="E8:80"/><i><font color="Green">E51</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="domain_1.html#K7">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, , <a class="ref" href="topreal2.html#T5">TOPREAL2:5</a></i>;<br/>
<a NAME="E9:80"/>
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R2p</font> = <font color="Maroon">D</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> 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>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E11:80"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">D</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>;<br/>
<a NAME="E12:80"/>
<font color="Maroon">P2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="jordan5b.html#T14">JORDAN5B:14</a></i>;<br/>
<a NAME="E13:80"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">R2p</font>, <a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="borsuk_3.html#R2">are_homeomorphic</a> 
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E14:80"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>, <a href="borsuk_4.html#K1" target="_self">I(01)</a>  <a href="borsuk_3.html#R1">are_homeomorphic</a> 
 ;<br/>


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