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 Th9:
  X c= Y implies X (-) B c= Y (-) B & X (+) B c= Y (+) B
proof
  assume
A1: X c= Y;
  thus X (-) B c= Y (-) B
  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;
    B+p1 c= Y by A1,A3;
    hence thesis by A2;
  end;
  let p be object;
  assume p in X (+) B;
  then ex p1,p2 being Point of T st p = p1+p2 & p1 in X & p2 in B;
  hence thesis by A1;
end;
