theorem Th38:
  for T being UnContinuous TopaddGroup, a being Element of T, W being
  a_neighborhood of -a ex A being open a_neighborhood of a st -A c= W
proof
  let T be UnContinuous TopaddGroup, a be Element of T,
W be a_neighborhood of -a;
  reconsider f = add_inverse T as Function of T, T;
  f.a = -a & f is continuous by Def7,Def6;
  then consider H being a_neighborhood of a such that
A1: f.:H c= W by BORSUK_1:def 1;
  a in Int Int H by CONNSP_2:def 1;
  then reconsider A = Int H as open a_neighborhood of a by CONNSP_2:def 1;
  take A;
  let x be object;
  assume x in -A;
  then consider g being Element of T such that
A2: x = -g and
A3: g in A;
  Int H c= H & f.g = -g by Def6,TOPS_1:16;
  then -g in f.:H by A3,FUNCT_2:35;
  hence thesis by A1,A2;
end;
