
theorem Th1:
  for A,B,C being non empty set, f being Function of A,B st for x
  being Element of A holds f.x in C holds f is Function of A,C
proof
  let A,B,C be non empty set, f be Function of A,B;
  assume for x being Element of A holds f.x in C;
  then dom f = A & for x be object holds x in A implies f.x in C
    by FUNCT_2:def 1;
  hence thesis by FUNCT_2:3;
end;
