<?xml version="1.0"?>
<div class="add">

<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">  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> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) &amp; <font color="Olive" title="b1">A</font> is  <a href="tsp_2.html#V1" title="TSP_2:attr.1">maximal_T_0</a>  holds <br/> not <font color="Olive" title="b1">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> </span><br/><span class="kw">let </span><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">X</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">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or <font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  ) &amp; <font color="Maroon" title="c2">A</font> is  <a href="tsp_2.html#V1" title="TSP_2:attr.1">maximal_T_0</a>  implies  not <font color="Maroon" title="c2">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:8"/>
( <font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or <font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">A</font> is  <a href="tsp_2.html#V1" title="TSP_2:attr.1">maximal_T_0</a>  or  not <font color="Maroon" title="c2">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a>  )</span><br/>

<a NAME="E2:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T56" title="TEX_4:th.56">TEX_4:56</a>, <a class="ref" href="tex_4.html#T60" title="TEX_4:th.60">TEX_4:60</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:8"/>
<font color="Maroon" title="c2">A</font> is  <a href="tsp_2.html#V1" title="TSP_2:attr.1">maximal_T_0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c2">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> </span><br/>

<a NAME="E4:8"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="tsp_2.html#D5" target="_self" title="TSP_2:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:8"/>
 not <font color="Maroon" title="c2">A</font> is  <a href="subset_1.html#V1" title="SUBSET_1:attr.1">proper</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#D6" title="SUBSET_1:def.6">SUBSET_1:def 6</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>
