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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x0</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="topgen_3.html#K6" target="_self">-PointClTop</a> <font color="Maroon">X</font></span>)</span>;<br/>





<a NAME="E1:56"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E2:56"/><b>then </b>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="topgen_3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:56"/><b>then </b>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x0</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>hence </b><a NAME="E4:56"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


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