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

<b>set </b><font color="Maroon">F</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E1:15_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:15_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font></span>)</span></span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E3:15_1_1"/>
( not <font color="Maroon">F</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">F</font> is <a href="taxonom2.html#V4">mutually-disjoint</a> )
 <b>by </b><i><a class="ref" href="taxonom2.html#T10">TAXONOM2:10</a></i>;<br/>

<b>thus </b><a NAME="E4:15_1_1"/>
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>thus </b><a NAME="E1:15_1_1_2"/>
(  not <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a><br/>

<b>thus </b><a NAME="E5:15_1_1"/>
(  not <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div>
