
theorem Th16:
  for X being non empty set, Y,Z being non empty Subset of ExtREAL
for F being Function of X,Y for G being Function of X,Z holds sup(F + G) <= sup
  F + sup G
proof
  let X be non empty set, Y,Z be non empty Subset of ExtREAL;
  let F be Function of X,Y;
  let G be Function of X,Z;
A1: sup(rng F + rng G) <= sup(rng F) + sup(rng G) by Th7;
  sup(F + G) <= sup(rng F + rng G) by Th15,XXREAL_2:59;
  hence thesis by A1,XXREAL_0:2;
end;
