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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#V2" title="TEX_3:attr.2">everywhere_dense</a>  &amp; <font color="Olive" title="b2">X2</font> is  <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  ) or ( <font color="Olive" title="b1">X1</font> is  <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  &amp; <font color="Olive" title="b2">X2</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  ) ) 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#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="c1">X</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( ( <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  &amp; <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  ) or ( <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  &amp; <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  ) ) 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#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="c1">X</font> )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">A1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">A2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><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></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:36"/>
( ( <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  &amp; <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  ) or ( <font color="Maroon" title="c2">X1</font> is  <a href="tex_3.html#V1" title="TEX_3:attr.1">dense</a>  &amp; <font color="Maroon" title="c3">X2</font> is  <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>  ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#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="c1">X</font></span><br/>

<a NAME="E5:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A1</font></span>: 
( ( <font color="Maroon" title="c4">A1</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  &amp; <font color="Maroon" title="c5">A2</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  ) or ( <font color="Maroon" title="c4">A1</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Maroon" title="c5">A2</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_3.html#T9" target="_self" title="TEX_3:th.9">Th9</a>, <a class="ref" href="tex_3.html#T16" target="_self" title="TEX_3:th.16">Th16</a></span>;<br/>
<a NAME="E6:36"/><span class="kw">then </span>
<font color="Maroon" title="c4">A1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="tex_3.html#E3"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a></span>;<br/>
<a NAME="E7:36"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#D3" title="TSEP_1:def.3">TSEP_1:def 3</a></span>;<br/>
<a NAME="E8:36"/>
<font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">A2</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="tops_3.html#T45" title="TOPS_3:th.45">TOPS_3:45</a></span>;<br/>
<a NAME="E9:36"/><span class="kw">then </span>
<font color="Maroon" title="c6">A</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="tsep_1.html#D5" title="TSEP_1:def.5">TSEP_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:36"/>
<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#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="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_3.html#T9" target="_self" title="TEX_3:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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