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<b>let </b><font color="Maroon">X0</font> be  non <a href="struct_0.html#V3">empty</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">X0</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:116_1_1"/>
<font color="Maroon">X0</font> is <a href="tex_2.html#V5" target="_self">maximal_discrete</a>
 ;<br/>

<a NAME="E3:116_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">A</font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 
<b>by </b><i><a class="ref" href="tex_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>assume </b><a NAME="E4:116_1_1"/>
<font color="Maroon">X0</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<a NAME="E5:116_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X0</font><b> such that </b><br/><a NAME="E7:116_1_1"/><i><font color="Green">E22</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E8:116_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">s</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E10:116_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:116_1_1"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tex_2.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E12:116_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tops_3.html#D2">TOPS_3:def 2</a></i>;<br/>
<a NAME="E13:116_1_1"/><b>then </b>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><br/> for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> holds <br/> <a href="pre_topc.html#K4">Cl</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tex_2.html#D4" target="_self">Def4</a>, </i>;<br/>
<b>hence </b><a NAME="E14:116_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tdlat_3.html#T22">TDLAT_3:22</a></i>;<br/>


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