let S, T be complete Scott TopLattice; :: thesis: for f being Function of S,T st S is algebraic & T is algebraic holds
( f is continuous iff for x being Element of S
for k being Element of T st k in the carrier of () holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of () & j <= x & k <= f . j ) ) )

let f be Function of S,T; :: thesis: ( S is algebraic & T is algebraic implies ( f is continuous iff for x being Element of S
for k being Element of T st k in the carrier of () holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of () & j <= x & k <= f . j ) ) ) )

assume that
A1: S is algebraic and
A2: T is algebraic ; :: thesis: ( f is continuous iff for x being Element of S
for k being Element of T st k in the carrier of () holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of () & j <= x & k <= f . j ) ) )

A3: S is continuous by ;
A4: T is continuous by ;
hereby :: thesis: ( ( for x being Element of S
for k being Element of T st k in the carrier of () holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of () & j <= x & k <= f . j ) ) ) implies f is continuous )
assume f is continuous ; :: thesis: for x being Element of S
for k being Element of T st k in the carrier of () holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of () & j <= x & k <= f . j ) )

then 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 A3, A4, Th23;
hence for x being Element of S
for k being Element of T st k in the carrier of () holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of () & j <= x & k <= f . j ) ) by A1, A2, Lm17; :: thesis: verum
end;
assume for x being Element of S
for k being Element of T st k in the carrier of () holds
( k <= f . x iff ex j being Element of S st
( j in the carrier of () & j <= x & k <= f . j ) ) ; :: thesis: f is continuous
then 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 ;
hence f is continuous by A3, A4, Th23; :: thesis: verum