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
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) ) )
let f be Function of S,T; :: thesis: ( f is continuous iff for x being Element of S
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) ) )

hereby :: thesis: ( ( for x being Element of S
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) ) ) implies f is continuous )

assume f is continuous ; :: thesis: for x being Element of S
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) )
then for x being Element of S holds f . x = "\/" ( { (f . w) where w is Element of S : w << x } ,T) by Th12;
hence for x being Element of S
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) ) by Th14; :: thesis: verum

end;

thus ( ( for x being Element of S
for y being Element of T holds
( y << f . x iff ex w being Element of S st
( w << x & y << f . w ) ) ) implies f is continuous ) by Lm15; :: thesis: verum