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<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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</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">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/> <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">F</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> ) </span>}</span> ;<br/>
<a NAME="E1:86"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T34" target="_self" title="TEX_4:th.34">Th34</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:86"/>
<font color="Maroon" title="c2">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font></span><br/>

<a NAME="E3:86"/><span class="kw">then </span>
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> ) </span>}</span> 
 
;<br/>
<a NAME="E4:86"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A2</font></span>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></span>;<br/>
<a NAME="E5:86"/>
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T57" target="_self" title="TEX_4:th.57">Th57</a></span>;<br/>
<a NAME="E6:86"/><span class="kw">then </span>
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:86"/>
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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