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

<b>assume </b><a NAME="E1:4"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:4"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan6.html#T66">JORDAN6:66</a></i>;<br/>
<a NAME="E3:4"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E4:4"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E5:4"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/>
<a NAME="E6:4"/><b>then </b>
 ex <font color="Olive">p3</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> st <br/>( <font color="Olive">p3</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Olive">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> &amp; <font color="Olive">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T55">JORDAN6:55</a></i>;<br/>
<b>hence </b><a NAME="E7:4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


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