theorem Th51:
  for G being BinContinuous TopaddGroup, A, O being Subset of G st O
  is open holds O + A is open
proof
  let G be BinContinuous TopaddGroup, A, O be Subset of G such that
A1: O is open;
  Int (O + A) = O + A
  proof
    thus Int (O + A) c= O + A by TOPS_1:16;
    let x be object;
    assume x in O + A;
    then consider o, a being Element of G such that
A2: x = o + a & o in O and
A3: a in A;
    set Q = O + a;
A4: Q c= O + A
    proof
      let q be object;
      assume q in Q;
      then ex h being Element of G st q = h + a & h in O by Th28;
      hence thesis by A3;
    end;
    x in Q by A2,Th28;
    hence thesis by A1,A4,TOPS_1:22;
  end;
  hence thesis;
end;
