
theorem Th41:
  for A being TopSpace, B being non empty TopSpace, f being
Function of A, B, C being TopSpace, X being Subset of A st f is continuous & C
  is SubSpace of B holds for h being Function of A|X, C st h = f|X holds h is
  continuous
proof
  let A be TopSpace, B be non empty TopSpace;
  let f be Function of A,B;
  let C be TopSpace, X be Subset of A;
  assume that
A1: f is continuous and
A2: C is SubSpace of B;
  the carrier of A|X = X by PRE_TOPC:8;
  then reconsider g = f|X as Function of A|X, B by FUNCT_2:32;
  let h be Function of A|X, C;
  assume
A3: h = f|X;
  g is continuous by A1,TOPMETR:7;
  hence thesis by A2,A3,PRE_TOPC:27;
end;
