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<div class="add">

<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1" target="_self">TopStruct</a> ;<br/><b>let </b><font color="Maroon">X'</font> be    <a href="pre_topc.html#M1" target="_self">SubSpace</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font>;<br/>







<b>assume </b><a NAME="E1:43"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:43"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A1</font> iff <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</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">X'</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:43_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A1</font> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</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">X'</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E2:43_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">D1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">D1</font> is <a href="pre_topc.html#V4" target="_self">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">D1</font> holds <br/><font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:43_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">D1</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 ;<br/>

<b>assume </b><a NAME="E2:43_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D1</font>
 ;<br/>

<a NAME="E3:43_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D1</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">X'</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<b>set </b><font color="Maroon">D3</font> = <font color="Maroon">D1</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">X'</font></span>)</span>;<br/>
<a NAME="E4:43_1_1_1"/>
<font color="Maroon">D1</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">X'</font></span>)</span> is <a href="pre_topc.html#V4" target="_self">closed</a>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E2:43_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="pre_topc.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:43_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</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">X'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E1:43"><i><font color="Green">E19</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E6:43_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">DD</font> =  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E4:43_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DD</font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E1:43"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E5:43_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="pre_topc.html#E1:43_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E6:43_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</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">X'</font></span>)</span>
 <b>by </b><i><a class="txt" href="pre_topc.html#E1:43"><i><font color="Green">E19</font></i></a>, <a class="txt" href="pre_topc.html#E2:43_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:43_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</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">X'</font></span>)</span>
 ;<br/>

<a NAME="E3:43_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">X'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">D1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font>;<br/><b>assume </b><a NAME="E1:43_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">D1</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 ;<br/><b>assume </b><a NAME="E2:43_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D1</font>
 ;<br/><b>consider </b><font color="Maroon">D2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:43_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">D2</font> is <a href="pre_topc.html#V4" target="_self">closed</a>
 <b>and </b><br/><a NAME="E5:43_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</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">X'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E6:43_1_2"/>
<font color="Maroon">D2</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">X'</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:43_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font>
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:43_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E9:43_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E1:43"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pre_topc.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="txt" href="pre_topc.html#E1:43"><i><font color="Green">E19</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:43"/>
 <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</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">X'</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
