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

<a NAME="E1:23_1_1"/>
<font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> is <a href="tsep_1.html#V1" target="_self">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tsep_1.html#D1" target="_self">TSEP_1:def 1</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 ;<br/>

<b>then </b><font color="Maroon">A</font> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span></span>)</span>

<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E3:23_1_1_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V1">strict</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tsep_1.html#V1" target="_self">open</a> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/>


</div>
