
theorem Th15:
  for X being non empty set, Y,Z being non empty Subset of ExtREAL,
     F being Function of X,Y, G being Function of X,Z holds
  rng(F + G) c= rng F + rng G
proof
  let X be non empty set, Y,Z be non empty Subset of ExtREAL, F be Function of
  X,Y, G be Function of X,Z;
A1: for x being object st x in X holds (F + G).x in rng F + rng G
  proof
    let x be object;
    assume x in X;
    then reconsider x as Element of X;
    reconsider a = F.x, b = G.x as R_eal;
A2: a in rng F by FUNCT_2:4;
A3: b in rng G by FUNCT_2:4;
    (F + G).x = a + b by Def3;
    hence thesis by A2,A3;
  end;
  dom (F + G) = X by FUNCT_2:def 1;
  then F + G is Function of X,rng F + rng G by A1,FUNCT_2:3;
  hence thesis by RELAT_1:def 19;
end;
