
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, k being Element of T st
  k in the carrier of CompactSublatt T
  holds (k <= f.x iff ex j being Element of S st
  j in the carrier of CompactSublatt S & j <= x & k <= f.j) )
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, 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, k being Element of T st
    k in the carrier of CompactSublatt T
    holds (k <= f.x iff ex j being Element of S st
    j in the carrier of CompactSublatt S & j <= x & k <= f.j) by A1,A2,Lm17;
  end;
  assume for x being Element of S, k being Element of T st
  k in the carrier of CompactSublatt T
  holds (k <= f.x iff ex j being Element of S st
  j in the carrier of CompactSublatt S & j <= x & k <= f.j);
  then 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 A2,Lm18;
  hence thesis by A3,A4,Th23;
end;
