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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"> for <font color="Olive" title="b1">A0</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>  st <font color="Olive" title="b1">A0</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> holds <br/> ex <font color="Olive" title="b2">X0</font> being <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</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="b2">X0</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> &amp; <font color="Olive" title="b1">A0</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="Olive" title="b2">X0</font> )</span><br/><b>let </b><font color="Maroon" title="c2">A0</font> be  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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A0</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> implies  ex <font color="Olive" title="b1">X0</font> being <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</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#V4" title="TEX_3:attr.4">nowhere_dense</a> &amp; <font color="Maroon" title="c2">A0</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="Olive" title="b1">X0</font> ) )</span><br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">A0</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</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 <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</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#V4" title="TEX_3:attr.4">nowhere_dense</a> &amp; <font color="Maroon" title="c2">A0</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="Olive" title="b1">X0</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c3">X0</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#V1" title="PRE_TOPC:attr.1">strict</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><br/><a NAME="E3:47"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c2">A0</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="c3">X0</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T10" title="TSEP_1:th.10">TSEP_1:10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">X0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X0</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> &amp; <font color="Maroon" title="c2">A0</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="c3">X0</font> )</span><br/>

<b>thus </b><a NAME="E4:47"/>
( <font color="Maroon" title="c3">X0</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> &amp; <font color="Maroon" title="c2">A0</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="c3">X0</font> )
 <b>by </b><i><a class="txt" href="#E1:47"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E3:47"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="tex_3.html#T35" target="_self" title="TEX_3:th.35">Th35</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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