 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;

theorem Th3:
for X be non empty TopSpace, S be non empty LinearTopSpace,
     f,g be Function of X,S
  st f is continuous & g is continuous
holds f+g is continuous
proof
  let X be non empty TopSpace, S be non empty LinearTopSpace,
      f,g be Function of X,S;
  assume A1: f is continuous & g is continuous;
  for x being Point of X holds (f+g) is_continuous_at x
  proof
    let x be Point of X;
    f is_continuous_at x & g is_continuous_at x by A1,TMAP_1:44;
    hence (f+g) is_continuous_at x by Th1;
  end;
  hence thesis by TMAP_1:44;
end;
