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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K3" title="JORDAN19:func.3">North_Arc</a> <font color="Maroon" title="c1">C</font> holds <br/> not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font></span><br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K3" title="JORDAN19:func.3">North_Arc</a> <font color="Maroon" title="c1">C</font> implies  not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font> )</span><br/>



<a NAME="E1:31"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="sprect_1.html#T33" title="SPRECT_1:th.33">SPRECT_1:33</a></i>;<br/>
<b>assume </b><a NAME="E2:31"/>
<font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K3" title="JORDAN19:func.3">North_Arc</a> <font color="Maroon" title="c1">C</font> or  not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font> )</span><br/>

<a NAME="E3:31"/><b>then </b>
(  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font> )
 
<b>by </b><i><a class="txt" href="jordan19.html#E1:31"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xreal_1.html#T228" title="XREAL_1:th.228">XREAL_1:228</a></i>;<br/>
<b>hence </b><a NAME="E4:31"/>
(  not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K3" title="JORDAN19:func.3">North_Arc</a> <font color="Maroon" title="c1">C</font> or  not <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="ref" href="jordan19.html#T27" target="_self" title="JORDAN19:th.27">Th27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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