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<div class="add">

<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> is <a href="tops_2.html#V2">closed</a> </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">ct</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="tarski.html#K1">}</a></span> as    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E2:21_1_1"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ct</font> holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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">T</font>;<br/>

<b>assume </b><a NAME="E1:21_1_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ct</font>
 ;<br/>

<a NAME="E2:21_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:21_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
;<br/>
<b>hence </b><a NAME="E4:21_1_1_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:21_1_1"/><b>then </b>
<font color="Maroon">ct</font> is <a href="tops_2.html#V2">closed</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>
<a NAME="E4:21_1_1"/><b>then </b>
<font color="Maroon">ct</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> is <a href="tops_2.html#V2">closed</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:21_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> : <font color="Olive">A</font> is <a href="tops_2.html#V2">closed</a> </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="t_1topsp.html#NM2">Part-Family</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div>
