
theorem
  for S, T being complete Scott TopLattice, f being Function of S, T holds
  S is algebraic & T is algebraic implies ( f is continuous iff
  for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w <= x & w is compact },T ) )
proof
  let S, T be complete Scott TopLattice, f be Function of S, T;
  assume that
A1: S is algebraic and
A2: T is algebraic;
A3: S is continuous by A1,WAYBEL_8:7;
A4: T is continuous by A2,WAYBEL_8:7;
  hereby
    assume f is continuous;
    then for x being Element of S holds
    f.x = "\/"({ f.w where w is Element of S : w << x },T) by A3,A4,Th24;
    hence for x being Element of S holds
    f.x = "\/"({ f.w where w is Element of S : w <= x & w is compact },T )
    by A1,A2,Lm19;
  end;
  assume for x being Element of S holds f.x = "\/"
  ({ f.w where w is Element of S : w <= x & w is compact },T );
  then for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w << x },T) by Th26;
  hence thesis by A3,A4,Th24;
end;
