<?xml version="1.0"?>
<div class="add">

<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">X1</font>, <font color="Olive" title="b2">X2</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 ( <font color="Olive" title="b1">X1</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> or <font color="Olive" title="b2">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> ) &amp; <font color="Olive" title="b1">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Olive" title="b2">X2</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Olive" title="b2">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a></span><br/><b>let </b><font color="Maroon" title="c2">X1</font>, <font color="Maroon" title="c3">X2</font> be  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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> or <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> ) &amp; <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> implies <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> )</span><br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> or <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</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">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</font> or <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> )</span><br/>

<b>assume </b><a NAME="E2:57"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#NR2" title="TSEP_1:NR.2">meets</a> <font color="Maroon" title="c3">X2</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">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a></span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1"/>
( <font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> or <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a> )
 </span><b>by </b><i><a class="txt" href="tex_3.html#E1:57"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_1_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c2">X1</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a></span><br/><div class="add"><a NAME="E2:57_1_1_1"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c2">X1</font>
 <b>by </b><i><a class="txt" href="tex_3.html#E2:57"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="tsep_1.html#T30" title="TSEP_1:th.30">TSEP_1:30</a></i>;<br/><b>hence </b><a NAME="E3:57_1_1_1"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E1:57_1_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="tex_3.html#T40" target="_self" title="TEX_3:th.40">Th40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_2"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a></span><br/><div class="add"><a NAME="E2:57_1_1_2"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c3">X2</font>
 <b>by </b><i><a class="txt" href="tex_3.html#E2:57"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="tsep_1.html#T30" title="TSEP_1:th.30">TSEP_1:30</a></i>;<br/><b>hence </b><a NAME="E3:57_1_1_2"/>
<font color="Maroon" title="c2">X1</font> <a href="tsep_1.html#K2" title="TSEP_1:func.2">meet</a> <font color="Maroon" title="c3">X2</font> is <a href="tex_3.html#V4" title="TEX_3:attr.4">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E1:57_1_1_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="tex_3.html#T40" target="_self" title="TEX_3:th.40">Th40</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>


</div>
