
theorem Th23:
  for S, T being continuous complete Scott TopLattice,
  f being Function of S, T holds ( f is continuous iff
  for x being Element of S, y being Element of T
  holds y << f.x iff ex w being Element of S st w << x & y << f.w )
proof
  let S, T be continuous complete Scott TopLattice, f be Function of S, T;
  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 Th12;
    hence for x being Element of S, y being Element of T
    holds y << f.x iff ex w being Element of S st w << x & y << f.w by Th14;
  end;
  thus thesis by Lm15;
end;
