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<b>set </b><font color="Maroon" title="c2">F</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">ct</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="eqrel_1.html#T48" title="EQREL_1:th.48">EQREL_1:48</a></i>;<br/>
<a NAME="E2:5_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ct</font> holds <br/><font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">ct</font>
 ;<br/>

<a NAME="E2:5_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1_1_1"/>
<font color="Maroon" title="c4">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:5_1_1"/><b>then </b>
<font color="Maroon" title="c3">ct</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a></i>;<br/>
<a NAME="E4:5_1_1"/><b>then </b>
<font color="Maroon" title="c3">ct</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E5:5_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> </span>}</span>  is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="eqrel_1.html#NM5" title="EQREL_1:NM.5">Part-Family</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="t_1topsp.html#T11" target="_self" title="T_1TOPSP:th.11">Th11</a></i>;<br/>


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