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
  for T being TopGroup st for a, b being Element of T, W being
a_neighborhood of a*(b") ex A being a_neighborhood of a, B being a_neighborhood
  of b st A*(B") c= W holds T is TopologicalGroup
proof
  let T be TopGroup such that
A1: for a, b being Element of T, W being a_neighborhood of a*(b") ex A
  being a_neighborhood of a, B being a_neighborhood of b st A*(B") c= W;
A2: for a being Element of T, W being a_neighborhood of a" ex A being
  a_neighborhood of a st A" c= W
  proof
    let a be Element of T, W be a_neighborhood of a";
    W is a_neighborhood of 1_T*(a") by GROUP_1:def 4;
    then consider
    A being a_neighborhood of 1_T, B being a_neighborhood of a such
    that
A3: A*(B") c= W by A1;
    take B;
    let x be object;
    assume
A4: x in B";
    then consider g being Element of T such that
A5: x = g" and
    g in B;
    1_T in A by CONNSP_2:4;
    then 1_T * (g") in A*(B") by A4,A5;
    then g" in A*(B") by GROUP_1:def 4;
    hence thesis by A3,A5;
  end;
  for a, b being Element of T, W being a_neighborhood of a*b ex A being
  a_neighborhood of a, B being a_neighborhood of b st A*B c= W
  proof
    let a, b be Element of T, W be a_neighborhood of a*b;
    W is a_neighborhood of a*(b"");
    then consider
    A being a_neighborhood of a, B being a_neighborhood of b" such
    that
A6: A*(B") c= W by A1;
    consider B1 being a_neighborhood of b such that
A7: B1" c= B by A2;
    take A, B1;
    let x be object;
    assume x in A * B1;
    then consider g, h being Element of T such that
A8: x = g * h & g in A and
A9: h in B1;
    h" in B1" by A9;
    then h in B" by A7,Th7;
    then x in A*(B") by A8;
    hence thesis by A6;
  end;
  hence thesis by A2,Th37,Th39;
end;
