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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">S</font> is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>  &amp; not <font color="Maroon" title="c1">S</font> is  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">p1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E2:26_1_1"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c2">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">p1</font> = <font color="Maroon" title="c2">p1</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>;<br/>
<a NAME="E4:26_1_1"/>
 ex <font color="Olive" title="b1">p2</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 <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal5.html#T21" target="_self" title="TOPREAL5:th.21">Th21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:26_1_1"/>
not <font color="Maroon" title="c1">S</font> is  <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="sppol_1.html#D3" title="SPPOL_1:def.3">SPPOL_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">S</font> is  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> </span><br/>

<a NAME="E6:26_1_1"/>
 ex <font color="Olive" title="b1">p2</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 <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal5.html#T20" target="_self" title="TOPREAL5:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:26_1_1"/>
not <font color="Maroon" title="c1">S</font> is  <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="sppol_1.html#D2" title="SPPOL_1:def.2">SPPOL_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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