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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> non <a href="realset2.html#V3">trivial</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="tex_2.html#V2">proper</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> holds not <font color="Olive">b<sub>1</sub></font> is <a href="tex_3.html#V1" target="_self">dense</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><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/><b>assume </b><a NAME="E1:59_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_1_1_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T17">TOPS_3:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:59_1_1_1_2"/><i><font color="Green">E29</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>3</sub></font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T10">TSEP_1:10</a></i>;<br/><a NAME="E4:59_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V1">proper</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E1:59_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="pre_topc.html#V1">strict</a> <a href="tex_2.html#V2">proper</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tex_3.html#E3:59_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tex_2.html#T13">TEX_2:13</a></i>;<br/><a NAME="E6:59_1_1_1_2"/>
not <font color="Maroon">c<sub>4</sub></font> is <a href="tex_3.html#V1" target="_self">dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E1:59"><i><font color="Green">E27</font></i></a></i>;<br/><b>hence </b><a NAME="E7:59_1_1_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E3:59_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tex_3.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:59_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:59"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V1">discrete</a>
 <b>by </b><i><a class="ref" href="tex_1.html#T35">TEX_1:35</a></i>;<br/>


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