
theorem Th11:
  for f being complex-valued Function holds dom (f^2) = dom f &
  for c being object holds f^2.c = (f.c)^2
proof
  let f be complex-valued Function;
  thus
A1: dom (f^2) = dom f /\ dom f by Def4
    .= dom f;
  let c be object;
  per cases;
  suppose
    c in dom f;
    hence thesis by A1,Def4;
  end;
  suppose
A2: not c in dom f;
    hence f^2.c = 0 qua Complex^2 by A1,FUNCT_1:def 2
      .= (f.c)^2 by A2,FUNCT_1:def 2;
  end;
end;
