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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</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">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/> <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Olive" title="b2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font></span><br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>; <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">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Maroon" title="c2">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> holds <br/> <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</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">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> implies  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> )</span><br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font></span><br/>

<a NAME="E2:39"/>
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">x</font> is <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#T22" target="_self" title="TEX_4:th.22">Th22</a></i>;<br/>
<a NAME="E3:39"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">x</font> is <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a>
 
<b>by </b><i><a class="ref" href="tex_4.html#D7" target="_self" title="TEX_4:def.7">Def7</a></i>;<br/>
<a NAME="E4:39"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T14" target="_self" title="TEX_4:th.14">Th14</a></i>;<br/>
<a NAME="E5:39"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E6:39"/>
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="tex_4.html#D2" target="_self" title="TEX_4:def.2">Def2</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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