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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="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> iff (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">regular_open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">regular_closed</a> &amp;  <a href="isomichi.html#K1" title="ISOMICHI:func.1">Border</a> <font color="Olive" title="b1">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</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="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> iff (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">regular_open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">regular_closed</a> &amp;  <a href="isomichi.html#K1" title="ISOMICHI:func.1">Border</a> <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )</span><br/>



<b>thus </b><a NAME="E1:47"/>
( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> implies (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">regular_open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">regular_closed</a> &amp;  <a href="isomichi.html#K1" title="ISOMICHI:func.1">Border</a> <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">regular_open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">regular_closed</a> &amp;  <a href="isomichi.html#K1" title="ISOMICHI:func.1">Border</a> <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">regular_open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">regular_closed</a> &amp;  <a href="isomichi.html#K1" title="ISOMICHI:func.1">Border</a> <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )</span><br/>

<a NAME="E2:47_1"/><b>then </b>
( <font color="Maroon" title="c2">A</font> is <a href="isomichi.html#V1" title="ISOMICHI:attr.1">supercondensed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="isomichi.html#V2" title="ISOMICHI:attr.2">subcondensed</a> )
 
<b>by </b><i><a class="ref" href="isomichi.html#T5" target="_self" title="ISOMICHI:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">regular_open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">regular_closed</a> &amp;  <a href="isomichi.html#K1" title="ISOMICHI:func.1">Border</a> <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 <b>by </b><i><a class="ref" href="isomichi.html#T25" target="_self" title="ISOMICHI:th.25">Th25</a>, <a class="ref" href="isomichi.html#T26" target="_self" title="ISOMICHI:th.26">Th26</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="E2:47"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">regular_open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">regular_closed</a> &amp;  <a href="isomichi.html#K1" title="ISOMICHI:func.1">Border</a> <font color="Maroon" title="c2">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a></span><br/>

<a NAME="E3:47"/><b>then </b>
( <font color="Maroon" title="c2">A</font> is <a href="isomichi.html#V1" title="ISOMICHI:attr.1">supercondensed</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="isomichi.html#V2" title="ISOMICHI:attr.2">subcondensed</a> )
 
<b>by </b><i><a class="ref" href="isomichi.html#T25" target="_self" title="ISOMICHI:th.25">Th25</a>, <a class="ref" href="isomichi.html#T26" target="_self" title="ISOMICHI:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V4" title="TOPS_1:attr.4">condensed</a>
 <b>by </b><i><a class="ref" href="isomichi.html#T5" target="_self" title="ISOMICHI:th.5">Th5</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>
