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<span class="kw">let </span><font color="Maroon" title="c1">TS</font> be   <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">  for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>  st <font color="Olive" title="b1">P</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Olive" title="b2">Q</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Olive" title="b2">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/><font color="Olive" title="b1">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">Q</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</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="c2">P</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Maroon" title="c3">Q</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  &amp; <font color="Maroon" title="c3">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c2">P</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> 
 <span class="kw">and </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TS</font>
 <span class="kw">and </span><br/><a NAME="E3:61"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c3">Q</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tops_1.html#D3" target="_self" title="TOPS_1:def.3">TOPS_1:def 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="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font> is  <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> </span><br/>

<span class="kw">thus </span><a NAME="E4:61"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">TS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:61"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="tops_1.html#T46" target="_self" title="TOPS_1:th.46">Th81</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tops_1.html#D3" target="_self" title="TOPS_1:def.3">TOPS_1:def 3</a> <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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