<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tex_2.html#E17"><i><font color="Green">Lemma9</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:116_1_1"/>
<font color="Maroon">c<sub>2</sub></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">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 
<b>by </b><i><a class="ref" href="tex_2.html#D8" target="_self">Def8</a></i>;<br/>
<b>assume </b><a NAME="E4:116_1_1"/>
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:116_1_1"/><i><font color="Green">E55</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E8:116_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E10:116_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tex_2.html#E7:116_1_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E11:116_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="txt" href="tex_2.html#E3:116_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="tex_2.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E12:116_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> holds <br/> <a href="pre_topc.html#K4">Cl</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tex_2.html#E10:116_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tex_2.html#T55" target="_self">Th55</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/>


</div>
