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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V1">discrete</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:91_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 ;<br/><a NAME="E2:91_1"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/><b>then 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>1</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/><a NAME="E4:91_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V3" target="_self">discrete</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="tex_2.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E5:91_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tex_2.html#E1:91_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tex_2.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E6:91_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V1" target="_self">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T5" target="_self">Th5</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:91_2"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V1" target="_self">proper</a>
 ;<br/><a NAME="E2:91_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</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="tex_2.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:91_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E4:91_2"/>
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> is <a href="tex_2.html#V3" target="_self">discrete</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="tex_2.html#E2:91_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E5:91_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 <b>by </b><i><a class="txt" href="tex_2.html#E3:91_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tex_2.html#D7" target="_self">Def7</a></i>;<br/></div>
<b>end;</b></div>


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