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<b>let </b><font color="Maroon" title="c1">T</font> be   <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">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="topgen_1.html#V5" title="TOPGEN_1:attr.5">perfect</a> iff  <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</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">A</font> is <a href="topgen_1.html#V5" title="TOPGEN_1:attr.5">perfect</a> iff  <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )</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"> (  <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c2">A</font> is <a href="topgen_1.html#V5" title="TOPGEN_1:attr.5">perfect</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon" title="c2">A</font> is <a href="topgen_1.html#V5" title="TOPGEN_1:attr.5">perfect</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font></span><br/><a NAME="E2:63_1"/><b>then </b>
( <font color="Maroon" title="c2">A</font> is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="ref" href="topgen_1.html#D10" target="_self" title="TOPGEN_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:63_1"/>
 <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="topgen_1.html#E54"><i><font color="Green" title="E34">Lm2</font></i></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>

<b>assume </b><a NAME="E2:63"/><i><font color="Green" title="E36">A1</font></i>: 
 <a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</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">A</font> is <a href="topgen_1.html#V5" title="TOPGEN_1:attr.5">perfect</a></span><br/>

<a NAME="E3:63"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="topgen_1.html#V2" title="TOPGEN_1:attr.2">dense-in-itself</a>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D7" target="_self" title="TOPGEN_1:def.7">Def7</a></i>;<br/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> = 
<span class="p1">(<span class="default"><a href="topgen_1.html#K2" title="TOPGEN_1:func.2">Der</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="topgen_1.html#T31" target="_self" title="TOPGEN_1:th.31">Th31</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="txt" href="topgen_1.html#E2:63"><i><font color="Green" title="E36">A1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:63"/>
<font color="Maroon" title="c2">A</font> is <a href="topgen_1.html#V5" title="TOPGEN_1:attr.5">perfect</a>
 <b>by </b><i><a class="txt" href="topgen_1.html#E3:63"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="topgen_1.html#D10" target="_self" title="TOPGEN_1:def.10">Def10</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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