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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X0</font> be   <a href="tsp_2.html#NM1">maximal_Kolmogorov_subspace</a> of <font color="Maroon">X</font>;<br/>



<b>reconsider </b><font color="Maroon">M</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E2:27"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">M</font> is <a href="tsp_2.html#V1" target="_self">maximal_T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">F0</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font>;<br/>



<b>assume </b><a NAME="E3:27"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 ;<br/>

<b>thus </b><a NAME="E4:27"/>
( <font color="Maroon">F0</font> is <a href="pre_topc.html#V4">closed</a> implies (  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon">F0</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>then consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:27_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">H</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:27_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="tsp_1.html#D2">TSP_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E5:27_1"/>
 <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_2.html#E3:27_1"><i><font color="Green">E19</font></i></a>, , <a class="txt" href="tsp_2.html#E4:27_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="tsp_2.html#E5:27"><i><font color="Green">E22</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E6:27_1"/>
<font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E3:27_1"><i><font color="Green">E19</font></i></a>, , <a class="txt" href="tsp_2.html#E4:27_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="tsp_2.html#E5:27"><i><font color="Green">E22</font></i></a>, </i>;<br/>


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

<b>assume </b><a NAME="E5:27"/><i><font color="Green">E22</font></i>: 
 <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>assume </b><a NAME="E6:27"/>
<font color="Maroon">F0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/>

<b>hence </b><a NAME="E7:27"/>
<font color="Maroon">F0</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="tsp_1.html#D2">TSP_1:def 2</a></i>;<br/>


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