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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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>  iff  for <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 <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> holds <br/> not <font color="Olive" title="b1">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <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 <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> holds <br/> not <font color="Olive" title="b1">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  ) implies <font color="Maroon" title="c1">X</font> is  <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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 <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> holds <br/> not <font color="Olive" title="b1">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> </span><br/><span class="kw">assume </span><a NAME="E2:53_1"/>
 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_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:53_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 <span class="kw">and </span><br/><a NAME="E5:53_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="tops_3.html#T2" title="TOPS_3:th.2">TOPS_3:2</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">B</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> </span><br/><span class="kw">thus </span><a NAME="E2:53_1_1"/>
<font color="Maroon" title="c4">B</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="tops_3.html#T39" title="TOPS_3:th.39">TOPS_3:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:53_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="tex_1.html#D6" target="_self" title="TEX_1:def.6">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:53"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
 for <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 <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> holds <br/> not <font color="Olive" title="b1">A</font> is  <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is  <a href="tdlat_3.html#V3" title="TDLAT_3:attr.3">almost_discrete</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:53"/>
 not <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:53"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tex_1.html#D6" target="_self" title="TEX_1:def.6">Def7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )</span><br/><span class="kw">thus </span><a NAME="E1:53_2"/>
( <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_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="tops_3.html#T1" title="TOPS_3:th.1">TOPS_3:1</a>, <a class="ref" href="tops_3.html#T40" title="TOPS_3:th.40">TOPS_3:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:53"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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