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<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">n</font> <a href="jordan1h.html#R1">is_sufficiently_large_for</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green">E38</font></i>: 
not  <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<a NAME="E3:16"/><i><font color="Green">E43</font></i>: 
 <a href="jordan14.html#K1" target="_self">SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E44</font></i>:  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> = 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gobrd14.html#T47">GOBRD14:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gobrd14.html#T31">GOBRD14:31</a></i>
;<br/>
<a NAME="E5:16"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 4
 
<b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a></i>;<br/>
<a NAME="E6:16"/><b>then </b><i><font color="Green">E45</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
 <a href="jordan14.html#K1" target="_self">SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan13.html#D1">JORDAN13:def 1</a></i>;<br/>
<a NAME="E10:16"/><b>then </b>
 <a href="jordan14.html#K1" target="_self">SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:16"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<a NAME="E12:16"/><i><font color="Green">E54</font></i>: 
 <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/>
<a NAME="E13:16"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T31">JORDAN2C:31</a></i>;<br/>
<a NAME="E14:16"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:16"/><b>then </b>
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E16:16"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E17:16"/>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/>
<a NAME="E18:16"/><b>then </b>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font>, <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T41">TSEP_1:41</a></i>;<br/>
<a NAME="E19:16"/><b>then </b>
 <a href="jordan2c.html#K1">BDD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , , <a class="ref" href="connsp_1.html#T17">CONNSP_1:17</a></i>;<br/>
<a NAME="E20:16"/><b>then </b>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>hence </b><a NAME="E21:16"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/>


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