theorem Th54:
  for G being TopologicaladdGroup, a being Point of G, A being
  a_neighborhood of a + (-a) ex B being open a_neighborhood of a st
B + (-B) c= A
proof
  let G be TopologicaladdGroup, a be Point of G, A be
a_neighborhood of a + (-a);
  consider X, Y being open a_neighborhood of a such that
A1: X+ (-Y) c= A by Th40;
  reconsider B = X /\ Y as open a_neighborhood of a by CONNSP_2:2;
  take B;
  let x be object;
  assume x in B+ (-B);
  then consider g, h being Point of G such that
A2: x = g+h and
A3: g in B and
A4: h in -B;
  -h in B by A4,Th7;
  then -h in Y by XBOOLE_0:def 4;
  then
A5: h in -Y by Th7;
  g in X by A3,XBOOLE_0:def 4;
  then x in X + (-Y) by A2,A5;
  hence thesis by A1;
end;
