let L be Lattice; :: thesis: for S being non empty Subset of L holds
( S is Filter of L iff for p, q being Element of L holds
( ( p in S & q in S ) iff p "/\" q in S ) )
let S be non empty Subset of L; :: thesis: ( S is Filter of L iff for p, q being Element of L holds
( ( p in S & q in S ) iff p "/\" q in S ) )
thus ( S is Filter of L implies for p, q being Element of L holds
( ( p in S & q in S ) iff p "/\" q in S ) ) by LATTICES:6, LATTICES:def 23, LATTICES:def 24; :: thesis: ( ( for p, q being Element of L holds
( ( p in S & q in S ) iff p "/\" q in S ) ) implies S is Filter of L )
assume A1: for p, q being Element of L holds
( ( p in S & q in S ) iff p "/\" q in S ) ; :: thesis: S is Filter of L
S is final

proof

let p, q be Element of L; :: according to LATTICES:def 23 :: thesis: ( not p [= q or not p in S or q in S )
assume that
A2: p [= q and
A3: p in S ; :: thesis: q in S
p "/\" q = p by A2, LATTICES:4;
hence q in S by A1, A3; :: thesis: verum

end;

hence S is Filter of L by A1, LATTICES:def 24; :: thesis: verum