theorem
  A is stable iff A * A c= A
proof
  hereby
    assume A1: A is stable;
    for x being object st x in A * A holds x in A
    proof
      let x be object;
      assume x in A * A; then
      consider v,w be Element of M such that
      A2: x = v * w & v in A & w in A by GROUP_2:8;
      thus x in A by A1,A2;
    end;
    hence A * A c= A;
  end;
  assume A3: A * A c= A;
  for v,w being Element of M st v in A & w in A holds v*w in A
  proof
    let v,w be Element of M;
    assume v in A & w in A; then
    v*w in A * A by GROUP_2:8;
    hence v*w in A by A3;
  end;
  hence A is stable;
end;
