
theorem Th1:
  for X being non empty TopSpace
  for y being Complex
  for f being Function of the carrier of X,COMPLEX st f=X --> y
      holds f is continuous
proof
  let X be non empty TopSpace;
  let y be Complex;
  let f be Function of the carrier of X,COMPLEX such that
A1:   f = X --> y;
  set H = the carrier of X;
  set HX=[#] X;
  let P1 be Subset of COMPLEX such that P1 is closed;
  per cases;
  suppose y in P1;
    then f"P1 = HX by A1,FUNCOP_1:14;
    hence f"P1 is closed;
  end;
  suppose not y in P1;
    then f"P1 = {}X by A1,FUNCOP_1:16;
    hence f"P1 is closed;
  end;
end;
