theorem Th40:
  for T being TopologicalGroup, 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 TopologicalGroup, 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;
