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<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="pre_topc.html#M1" target="_self">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<a NAME="E1:39"/><i><font color="Green">E16</font></i>: 
 <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T39" target="_self">Th39</a></i>;<br/>
<div><i><font color="Green">E17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 ;<br/><a NAME="E2:39_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E3:39_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" target="_self">Def5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:39_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:39_1"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" target="_self">Def9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E8:39_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>by </b><i><a class="txt" href="pre_topc.html#E5:39_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pre_topc.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E9:39_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E10:39_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>by </b><i><a class="txt" href="pre_topc.html#E8:39_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="pre_topc.html#D6" target="_self">Def6</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="pre_topc.html#E1:39"><i><font color="Green">E16</font></i></a>, <a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="pre_topc.html#E6:39_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pre_topc.html#T22" target="_self">Th22</a></i>
;<br/><b>hence </b><a NAME="E12:39_1"/>
ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="pre_topc.html#E10:39_1"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:39_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>and </b><br/><a NAME="E2:39_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E3:39_2"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>by </b><i><a class="txt" href="pre_topc.html#E1:39_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pre_topc.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E4:39_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" target="_self">Def5</a></i>;<br/><i><font color="Green">E21</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="pre_topc.html#E2:39_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="pre_topc.html#E1:39"><i><font color="Green">E16</font></i></a>, <a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T17" target="_self">Th17</a></i>
;<br/><a NAME="E6:39_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1" target="_self">topology</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E4:39_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="pre_topc.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E7:39_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>by </b><i><a class="txt" href="pre_topc.html#E5:39_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pre_topc.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E8:39_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a> iff ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="pre_topc.html#E2:39"><i><font color="Green">E17</font></i></a></i>;<br/>


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