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 Th54:
  for G being TopologicalGroup, 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 TopologicalGroup, 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;
