
theorem Th21:
  for X, Y being non empty TopSpace, f being continuous Function
  of Omega X, Omega Y holds f is monotone
proof
  let X, Y be non empty TopSpace, f be continuous Function of Omega X, Omega Y;
  let x, y be Element of Omega X;
  reconsider Z = {f.y} as Subset of Y by Lm1;
  assume x <= y;
  then consider A being Subset of X such that
A1: A = {y} and
A2: x in Cl A by Def2;
A3: for G being Subset of Y st G is open holds f.x in G implies Z meets G
  proof
    the carrier of X = the carrier of Omega X by Lm1;
    then reconsider g = f as Function of X, Y by Lm1;
    let G be Subset of Y such that
A4: G is open and
A5: f.x in G;
A6: x in g"G by A5,FUNCT_2:38;
A7: the TopStruct of Y = the TopStruct of Omega Y by Def2;
A8: f.y in Z by TARSKI:def 1;
    the TopStruct of X = the TopStruct of Omega X by Def2;
    then
A9: g is continuous by A7,YELLOW12:36;
    [#]Y <> {};
    then g"G is open by A4,A9,TOPS_2:43;
    then A meets g"G by A2,A6,PRE_TOPC:def 7;
    then consider m being object such that
A10: m in A /\ g"G by XBOOLE_0:4;
    m in A by A10,XBOOLE_0:def 4;
    then
A11: m = y by A1,TARSKI:def 1;
    m in g"G by A10,XBOOLE_0:def 4;
    then f.y in G by A11,FUNCT_2:38;
    then Z /\ G <> {}Y by A8,XBOOLE_0:def 4;
    hence thesis;
  end;
  the carrier of Y = the carrier of Omega Y by Lm1;
  then f.x in Cl Z by A3,PRE_TOPC:def 7;
  hence f.x <= f.y by Def2;
end;
