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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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:64"/><i><font color="Green">E13</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) </span>}</span> ;<br/>
<a NAME="E2:64"/>
<span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</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="E1:64_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:64_1"/><b>then </b>
 ex <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> )
 
;<br/>
<b>hence </b><a NAME="E3:64_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">Y</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:64_2"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:64_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:64_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> )
 ;<br/><b>thus </b><a NAME="E5:64_2"/>
 <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">C</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T6" target="_self">Th6</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:64"/>
 <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">F</font> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">F</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tex_4.html#D4" target="_self">Def4</a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


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