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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st ( <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) &amp; <font color="Olive" title="b1">A</font> is <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a> holds <br/> not <font color="Olive" title="b1">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> ) &amp; <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a> implies  not <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> )</span><br/>



<b>assume </b><a NAME="E1:106"/>
( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a> or  not <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> )</span><br/>

<a NAME="E2:106"/><b>then </b><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T23" title="TDLAT_3:th.23">TDLAT_3:23</a>, <a class="ref" href="tdlat_3.html#T24" title="TDLAT_3:th.24">TDLAT_3:24</a></i>;<br/>
<a NAME="E3:106"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<b>assume </b><a NAME="E4:106"/>
<font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V4" title="TEX_2:attr.4">maximal_discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a></span><br/>

<a NAME="E5:106"/><b>then </b>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 
<b>by </b><i><a class="txt" href="tex_2.html#E2:106"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="tex_2.html#T47" target="_self" title="TEX_2:th.47">Th47</a></i>;<br/>
<a NAME="E6:106"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="tex_2.html#E3:106"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="tops_3.html#D2" title="TOPS_3:def.2">TOPS_3:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:106"/>
not <font color="Maroon" title="c2">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T5" target="_self" title="TEX_2:th.5">Th5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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