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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V3">almost_discrete</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y0</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V1">discrete</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>



<b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y0</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="E2:114"/>
<font color="Maroon">A</font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">M</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:114"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E5:114"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">M</font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 <b>by </b><i/>;<br/>
<a NAME="E6:114"/>
not <font color="Maroon">M</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="tex_2.html#T9" target="_self">Th9</a></i>;<br/>
<b>then consider </b><font color="Maroon">X0</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:114"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">X0</font> is <a href="tex_2.html#V5" target="_self">maximal_discrete</a>
 <b>and </b><br/><a NAME="E9:114"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 <b>by </b><i>, <a class="txt" href="tex_2.html#E17"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">X0</font>
;<br/>

<b>thus </b><a NAME="E10:114"/>
( <font color="Maroon">Y0</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X0</font> &amp; <font color="Maroon">X0</font> is <a href="tex_2.html#V5" target="_self">maximal_discrete</a> )
 <b>by </b><i><a class="ref" href="tex_2.html#T18" target="_self">Th18</a>, <a class="ref" href="tex_2.html#D4" target="_self">Def4</a>, , <a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/>


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