
theorem
  for C being set, D1,D2 being complex-membered non empty set for f1
  being Function of C,D1, f2 being Function of C,D2 for c being Element of C
  holds (f1+f2).c = f1.c + f2.c
proof
  let C be set;
  let D1,D2 be complex-membered non empty set;
  let f1 be Function of C,D1, f2 be Function of C,D2;
A1: dom(f1+f2) = C by FUNCT_2:def 1;
  let c be Element of C;
  per cases;
  suppose
    C is non empty;
    hence thesis by A1,Def1;
  end;
  suppose
A2: C is empty;
    then f1.c = 0;
    hence thesis by A2;
  end;
end;
