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 Th53:
  for G being UnContinuous TopGroup, a being Point of G, A being
  a_neighborhood of a holds A" is a_neighborhood of a"
proof
  let G be UnContinuous TopGroup, a be Point of G, A be a_neighborhood of a;
  a in Int A by CONNSP_2:def 1;
  then consider Q being Subset of G such that
A1: Q is open and
A2: Q c= A & a in Q by TOPS_1:22;
  Q" c= A" & a" in Q" by A2,Th8;
  hence a" in Int (A") by A1,TOPS_1:22;
end;
