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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:57"/>
( not <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> implies  ex <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 <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/>
not <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">A</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:57_1"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</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="tex_1.html#T37" target="_self" title="TEX_1:th.37">Th37</a></i>;<br/>
<b>take </b><font color="Maroon" title="c3">B</font> = <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>

<b>thus </b><a NAME="E4:57_1"/>
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">B</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="tex_1.html#E3:57_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a>, <a class="ref" href="tops_1.html#D4" title="TOPS_1:def.4">TOPS_1:def 4</a>, <a class="ref" href="tops_3.html#T1" title="TOPS_3:th.1">TOPS_3:1</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E2:57"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E3:57"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c3">B</font> = <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="tex_1.html#E2:57"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="tops_3.html#T2" title="TOPS_3:th.2">TOPS_3:2</a></i>;<br/><b>take </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c3">B</font>;<br/><b>thus </b><a NAME="E2:57_2"/>
( <font color="Maroon" title="c4">B</font> is <a href="tops_1.html#V2" title="TOPS_1:attr.2">boundary</a> &amp; <font color="Maroon" title="c4">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="tex_1.html#E3:57"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a>, <a class="ref" href="tops_3.html#T18" title="TOPS_3:th.18">TOPS_3:18</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:57"/>
not <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>
 <b>by </b><i><a class="ref" href="tex_1.html#T37" target="_self" title="TEX_1:th.37">Th37</a></i>;<br/>


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