
theorem Th4:
  for S, T being up-complete Scott TopLattice,
  f being Function of S, T holds f is continuous implies f is monotone
proof
  let S, T be up-complete Scott TopLattice;
  let f be Function of S, T;
  assume
A1: f is continuous;
  let x,y be Element of S;
A2: dom f = the carrier of S by FUNCT_2:def 1;
  assume
A3: x <= y;
  f.x <= f.y
  proof
    set V = (downarrow (f.y))`, U1 = f"V;
    assume not f.x <= f.y;
    then not f.x in downarrow (f.y) by WAYBEL_0:17;
    then
A4: f.x in V by XBOOLE_0:def 5;
    f.y <= f.y;
    then
A5: f.y in downarrow (f.y) by WAYBEL_0:17;
    downarrow (f.y) is closed by Lm3;
    then U1 is open by A1,TOPS_2:43;
    then reconsider U1 as upper Subset of S by WAYBEL11:def 4;
    x in U1 by A2,A4,FUNCT_1:def 7;
    then y in U1 by A3,WAYBEL_0:def 20;
    then f.y in V by FUNCT_1:def 7;
    hence contradiction by A5,XBOOLE_0:def 5;
  end;
  hence thesis;
end;
