
theorem
  for f1,f2 being complex-valued Function
   for c being object holds (f1(#)f2).c = f1.c * f2.c
proof
  let f1,f2 be complex-valued Function;
  let c be object;
A1: dom (f1(#)f2) = dom f1 /\ dom f2 by Def4;
  per cases;
  suppose
    c in dom (f1(#)f2);
    hence thesis by Def4;
  end;
  suppose
A2: not c in dom (f1(#)f2);
    then not c in dom f1 or not c in dom f2 by A1,XBOOLE_0:def 4;
    then f1.c = 0 or f2.c = 0 by FUNCT_1:def 2;
    hence thesis by A2,FUNCT_1:def 2;
  end;
end;
