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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">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:82"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E2:82"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<b>assume </b><a NAME="E3:82"/><i><font color="Green">E49</font></i>: 
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E4:82"/>
 <a href="jordan6.html#K8" target="_self">Upper_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/>;<br/>
<a NAME="E5:82"/><b>then </b>
 <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E11:82"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan6.html#E12:82"><i><font color="Green">E55</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:82"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E8:82"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="txt" href="jordan6.html#E13:82"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fs</font> = <font color="Maroon">x</font> as   <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>by </b><i><a class="txt" href="jordan6.html#E13:82"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E10:82"/>
 <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:82"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E11:82"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">fs</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">fs</font> <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/>
<i><font color="Green">E55</font></i>: <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  = 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">fs</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, </i>
;<br/>
<i><font color="Green">E56</font></i>: <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  = 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">fs</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, </i>
;<br/>
<a NAME="E14:82"/>
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E15:82"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, , , <a class="ref" href="goboard7.html#T8">GOBOARD7:8</a></i>;<br/>
<b>hence </b><a NAME="E16:82"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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