reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem Th39:
  for T being TopGroup st for a being Element of T, W being
  a_neighborhood of a" ex A being a_neighborhood of a st A" c= W holds T is
  UnContinuous
proof
  let T be TopGroup such that
A1: for a being Element of T, W being a_neighborhood of a" ex A being
  a_neighborhood of a st A" c= W;
  set f = inverse_op T;
  for W being Point of T, G being a_neighborhood of f.W ex H being
  a_neighborhood of W st f.:H c= G
  proof
    let a be Point of T, G be a_neighborhood of f.a;
    f.a = a" by GROUP_1:def 6;
    then consider A being a_neighborhood of a such that
A2: A" c= G by A1;
    take A;
    thus thesis by A2,Th9;
  end;
  hence f is continuous by BORSUK_1:def 1;
end;
