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

<b>set </b><font color="Maroon">u</font> =  <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">l</font> =  <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font>;<br/>
<a NAME="E1:50"/><i><font color="Green">E46</font></i>: 
 <a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:50"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E54</font></i>: 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 ;<br/><a NAME="E2:50_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i/>;<br/><a NAME="E3:50_1"/>
 <a href="jordan21.html#K1" target="_self">UMP</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:50_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan21.html#E3"><i><font color="Green">Lemma40</font></i></a>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
<span class="p1">(<span class="default"><a href="jordan21.html#K2" target="_self">LMP</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


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