reserve p1, p2 for Point of TOP-REAL 2,
  C for Simple_closed_curve,
  P for Subset of TOP-REAL 2;

theorem Th12:
  for T1,T2 being non empty TopSpace, f being Function of T1,T2, A
  being Subset of T1 holds f|A is Function of T1|A, T2|(f.:A)
proof
  let T1,T2 be non empty TopSpace, f be Function of T1,T2, A be Subset of T1;
A1: rng (f|A) = f.:A by RELAT_1:115;
  dom f = the carrier of T1 by FUNCT_2:def 1;
  then
A2: dom (f|A) = A by RELAT_1:62;
  [#](T1|A) = A & [#](T2|(f.:A)) = f.:A by PRE_TOPC:def 5;
  hence thesis by A2,A1,FUNCT_2:2;
end;
