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 Th52:
  for G being BinContinuous TopGroup, A, O being Subset of G st O
  is open holds A * O is open
proof
  let G be BinContinuous TopGroup, A, O be Subset of G such that
A1: O is open;
  Int (A * O) = A * O
  proof
    thus Int (A * O) c= A * O by TOPS_1:16;
    let x be object;
    assume x in A * O;
    then consider a, o being Element of G such that
A2: x = a * o and
A3: a in A and
A4: o in O;
    set Q = a * O;
A5: Q c= A * O
    proof
      let q be object;
      assume q in Q;
      then ex h being Element of G st q = a * h & h in O by GROUP_2:27;
      hence thesis by A3;
    end;
    x in Q by A2,A4,GROUP_2:27;
    hence thesis by A1,A5,TOPS_1:22;
  end;
  hence thesis;
end;
