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<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>





<b>assume </b><a NAME="E1:63"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>assume </b><a NAME="E2:63"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:63"/>
 <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E3:63_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tex_4.html#T6" target="_self">Th6</a></i>;<br/>
<b>then consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:63_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E6:63_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E8:63_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E9:63_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:63_1"/><i><font color="Green">E29</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="txt" href="tex_4.html#E8:63_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E11:63_1"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">a</font> &amp;  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="tex_4.html#D3" target="_self">Def3</a>, <a class="ref" href="tex_4.html#D4" target="_self">Def4</a>, , <a class="txt" href="tex_4.html#E1:63_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tex_4.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E12:63_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:63_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
 <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/>


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