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

proof

assume A1: X is filtered ; :: thesis: for Y being finite Subset of X st Y <> {} holds
"/\" Y,L in X
let Y be finite Subset of X; :: thesis: ( Y <> {} implies "/\" Y,L in X )
assume A2: Y <> {} ; :: thesis: "/\" Y,L in X
consider z being Element of L such that
A3: ( z in X & 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 A3, YELLOW_0:31;
hence "/\" Y,L in X by A3, Def20; :: thesis: verum

end;

assume A4: for Y being finite Subset of X st Y <> {} holds
"/\" Y,L in X ; :: thesis: X is filtered
consider x being Element of X;
reconsider x = x as Element of L ;

now

let Y be finite Subset of X; :: thesis: ex x being Element of L st
( b₂ in X & b₂ is_<=_than x )

per cases ( Y = {} or Y <> {} ) ;

suppose Y = {} ; :: thesis: ex x being Element of L st
( b₂ in X & b₂ is_<=_than x )

then x is_<=_than Y by YELLOW_0:6;
hence ex x being Element of L st
( x in X & x is_<=_than Y ) ; :: thesis: verum

end;

suppose A5: Y <> {} ; :: thesis: ex x being Element of L st
( b₂ in X & b₂ is_<=_than x )

Y c= the carrier of L by XBOOLE_1:1;
then ex_inf_of Y,L by A5, YELLOW_0:55;
then ( "/\" Y,L in X & "/\" Y,L is_<=_than Y ) by A4, A5, YELLOW_0:31;
hence ex x being Element of L st
( x in X & x is_<=_than Y ) ; :: thesis: verum

end;

end;

end;

hence X is filtered by Th2; :: thesis: verum