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<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>assume </b><a NAME="E1:99_3_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a>
 ;<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/>
<a NAME="E3:99_3_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="txt" href="tex_2.html#E1:99_3_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<a NAME="E4:99_3_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> st <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>2</sub></font> holds <br/><font color="Olive">b<sub>1</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#T50" target="_self">Th50</a></i>;<br/>
<b>hence </b><a NAME="E5:99_3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V5" target="_self">maximal_discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#D8" target="_self">Def8</a></i>;<br/>


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