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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="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"> (  ex <font color="Olive" title="b1">X0</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> &amp; <font color="Olive" title="b1">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> &amp; <font color="Olive" title="b1">X0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</a> ) implies  not <font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> )</span><br/>

<b>given </b><font color="Maroon" title="c2">X0</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E2:79"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a> &amp; <font color="Maroon" title="c2">X0</font> is <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> &amp; <font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</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="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a></span><br/>

<b>assume </b><a NAME="E3:79"/>
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c1">X</font> as  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> ;<br/>
<a NAME="E5:79"/>
<font color="Maroon" title="c2">X0</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a> <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="tex_3.html#E2:79"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:79"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_3.html#E2:79"><i><font color="Green" title="E27">A1</font></i></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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