theorem Th40:
  for T being TopologicaladdGroup, a, b being Element of T for W
  being a_neighborhood of a+ (-b) ex A being open a_neighborhood of a, B being
  open a_neighborhood of b st A + (-B) c= W
proof
  let T be TopologicaladdGroup, a, b be Element of T,
W be a_neighborhood of a+(-b);
  consider A being open a_neighborhood of a, B being open a_neighborhood of -b
  such that
A1: A+B c= W by Th36;
  consider C being open a_neighborhood of b such that
A2: -C c= B by Th38;
  take A, C;
  let x be object;
  assume x in A+(-C);
  then consider g, h being Element of T such that
A3: x = g+h and
A4: g in A and
A5: h in -C;
  consider k being Element of T such that
A6: h = -k and
  k in C by A5;
  g+(-k) in A+B by A2,A4,A5,A6;
  hence thesis by A1,A3,A6;
end;
