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<b>let </b><font color="Maroon" title="c1">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font>
 ;<br/>

<a NAME="E2:12"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : <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> <font color="Maroon" title="c1">s</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#D6" title="JORDAN6:def.6">JORDAN6:def 6</a></i>;<br/>
<a NAME="E3:12"/>
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><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> <font color="Olive" title="b2">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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>;<br/>

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>
<a NAME="E3:12_1"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
 ex <font color="Olive" title="b1">p1</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">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> )
 
<b>by </b><i><a class="txt" href="jordan1a.html#E2:12"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:12_1"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E5:12_1"/><b>then </b>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c1">s</font>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>
<a NAME="E6:12_1"/><b>then </b>
 ex <font color="Olive" title="b1">p1</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">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">s</font> )
 
<b>by </b><i><a class="txt" href="jordan1a.html#E2:12"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:12_1"/>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan1a.html#E3:12_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12"/>
<font color="Maroon" title="c2">A</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="ref" href="sppol_1.html#D3" title="SPPOL_1:def.3">SPPOL_1:def 3</a></i>;<br/>


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