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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> ) </span>}</span> ;<br/>
<a NAME="E1:83"/>
<span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</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">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_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="E1:83_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:83_1"/><b>then </b>
 ex <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</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#V3">open</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="E3:83_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">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:83"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E4:83"/>
 <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">G</font> where <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


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