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<b>let </b><font color="Maroon">T1</font> be  <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T2</font> be  <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48_1_2"/><i><font color="Green">E50</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:48_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> holds <br/> ( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> iff ( <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> or <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> ) )
 <b>and </b><br/><a NAME="E3:48_1_2"/><i><font color="Green">E60</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:48_1_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> holds <br/> ( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> iff ( <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> or <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:48_1_2_1"/>
( <font color="Maroon">F</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> iff ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; ( <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> or <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> ) ) )
 <b>by </b><i><a class="ref" href="topgen_3.html#T1" target="_self">Th1</a>, </i>;<br/><b>hence </b><a NAME="E2:48_1_2_1"/>
( <font color="Maroon">F</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> iff <font color="Maroon">F</font> is  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:48_1_2"/>
<font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font>
 <b>by </b><i/>;<br/>


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