let L be with_infima Poset; :: thesis:
let X be non empty upper Subset of L; :: thesis:
thus ( X is filtered implies for Y being finite Subset of X st Y <> {} holds
"/\" (Y,L) in X ) :: thesis:

proof

assume A1: X is filtered ; :: thesis:
let Y be finite Subset of X; :: thesis:
assume A2: Y <> {} ; :: thesis:
consider z being Element of L such that
A3: z in X and
A4: Y is_>=_than z by A1, Th2;
Y c= the carrier of L by XBOOLE_1:1;
then ex_inf_of Y,L by A2, YELLOW_0:55;
then "/\" (Y,L) >= z by A4, YELLOW_0:31;
hence "/\" (Y,L) in X by A3, Def20; :: thesis:

end;

assume A5: for Y being finite Subset of X st Y <> {} holds
"/\" (Y,L) in X ; :: thesis:
set x = the Element of X;
reconsider x = the Element of X as Element of L ;

now :: thesis:

let Y be finite Subset of X; :: thesis:

per cases ;

suppose Y = {} ; :: thesis:

then x is_<=_than Y ;
hence ex x being Element of L st
( x in X & x is_<=_than Y ) ; :: thesis:

end;

suppose A6: Y <> {} ; :: thesis:

Y c= the carrier of L by XBOOLE_1:1;
then ex_inf_of Y,L by A6, YELLOW_0:55;
then "/\" (Y,L) is_<=_than Y by YELLOW_0:31;
hence ex x being Element of L st
( x in X & x is_<=_than Y ) by A5, A6; :: thesis:

end;

end;

end;

hence X is filtered by Th2; :: thesis: