let S, T be complete continuous Scott TopLattice; :: thesis: for f being Function of S,T holds
( f is continuous iff for x being Element of S holds f . x = "\/" ( { (f . w) where w is Element of S : w << x } ,T) )
let f be Function of S,T; :: thesis: ( f is continuous iff for x being Element of S holds f . x = "\/" ( { (f . w) where w is Element of S : w << x } ,T) )
thus ( f is continuous implies for x being Element of S holds f . x = "\/" ( { (f . w) where w is Element of S : w << x } ,T) ) by Th12; :: thesis: ( ( for x being Element of S holds f . x = "\/" ( { (f . w) where w is Element of S : w << x } ,T) ) implies f is continuous )
assume for x being Element of S holds f . x = "\/" ( { (f . w) where w is Element of S : w << x } ,T) ; :: thesis: f is continuous
then f is directed-sups-preserving by Lm16;
hence f is continuous ; :: thesis: verum