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
  0.T in B implies X (-) B c= X & X c= X (+) B
proof
  assume
A1: 0.T in B;
  thus X (-) B c= X
  proof
    let p be object;
    assume p in X (-) B;
    then consider p1 being Point of T such that
A2: p = p1 and
A3: B+p1 c= X;
    0.T+p1 in {q+p1 where q is Point of T:q in B} by A1;
    then 0.T+p1 in X by A3;
    hence thesis by A2;
  end;
  let p be object;
  assume
A4: p in X;
  then reconsider p as Point of T;
  p + 0.T in {x+b where x,b is Point of T:x in X & b in B} by A1,A4;
  hence thesis;
end;
