<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>thus </b><a NAME="E1:25"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="sppol_1.html#V2">vertical</a> implies  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="sppol_1.html#V2">vertical</a>
 ;<br/>

<a NAME="E2:25_1"/><i><font color="Green">E19</font></i>: 
(  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="pscomp_1.html#K32">E-min</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T15" target="_self">Th15</a>, <a class="ref" href="sprect_1.html#T16" target="_self">Th16</a></i>;<br/>
<b>thus </b> <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K32">E-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="sprect_1.html#E1:25_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="sprect_1.html#E2:25_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a></i>
<br/>.= 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E18</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_1.html#D3">SPPOL_1:def 3</a><br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <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>;<br/>



<b>assume </b><a NAME="E3:25"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E4:25"/><b>then </b>
(  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E5:25"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <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<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:25"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E6:25"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> 
 ;<br/>


</div>
