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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1" target="_self">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:49"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a> implies  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 ;<br/>

<a NAME="E2:49_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E3:49_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E1:49_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#T45" target="_self">Th45</a></i>;<br/>
<a NAME="E4:49_1"/><b>then </b>
 <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:49_1"/>
 <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E2:49_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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

<b>assume </b><a NAME="E2:49"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V2" target="_self">TopSpace-like</a>
 ;<br/>

<b>assume </b><a NAME="E3:49"/><i><font color="Green">E24</font></i>: 
 <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:49"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>and </b><br/><a NAME="E6:49"/><i><font color="Green">E26</font></i>: 
 <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E2:49"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E7:49"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E5:49"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:49"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>by </b><i><a class="txt" href="pre_topc.html#E2:49"><i><font color="Green">E23</font></i></a>, <a class="txt" href="pre_topc.html#E3:49"><i><font color="Green">E24</font></i></a>, <a class="txt" href="pre_topc.html#E6:49"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pre_topc.html#T44" target="_self">Th44</a></i>;<br/>


</div>
