reserve T for non empty Abelian
  add-associative right_zeroed right_complementable RLSStruct,
  X,Y,Z,B,C,B1,B2 for Subset of T,
  x,y,p for Point of T;

theorem Th1:
  for T being add-associative right_zeroed right_complementable
    non empty RLSStruct,
       B being Subset of T holds
  (B!)! = B
proof
  let T be add-associative right_zeroed right_complementable
    non empty RLSStruct,
      B be Subset of T;
  thus (B!)! c= B
  proof
    let x be object;
    assume x in (B!)!;
    then consider q being Element of T such that
A1: x = -q and
A2: q in B!;
    ex q1 being Element of T st q = -q1 & q1 in B by A2;
    hence thesis by A1;
  end;
  let x be object;
  assume
A3: x in B;
  then reconsider xx = x as Point of T;
  -xx in {-q where q is Point of T : q in B}by A3;
  then -(-xx) in {-q where q is Point of T : q in B!};
  hence thesis;
end;
