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
  X (+) (Y+p) = (X (+) Y)+p
proof
  thus X (+) (Y+p) c= (X (+) Y)+p
  proof
    let x be object;
    assume x in X (+) (Y+p);
    then consider x2,y2 being Point of T such that
A1: x = x2+y2 & x2 in X and
A2: y2 in Y+p;
    consider y3 being Point of T such that
A3: y2 = y3+p & y3 in Y by A2;
    x=x2+y3+p & x2+y3 in X (+) Y by A1,A3,RLVECT_1:def 3;
    hence thesis;
  end;
  let x be object;
  assume x in (X (+) Y)+p;
  then consider x2 being Point of T such that
A4: x = x2+p and
A5: x2 in X (+) Y;
  consider x3,y3 being Point of T such that
A6: x2 = x3+y3 and
A7: x3 in X and
A8: y3 in Y by A5;
A9: y3+p in Y+p by A8;
  x=x3+(y3+p) by A4,A6,RLVECT_1:def 3;
  hence thesis by A7,A9;
end;
