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 Th15:
  (X+p) (+) Y = (X (+) Y)+p
proof
  thus (X+p) (+) Y c= (X (+) Y)+p
  proof
    let x be object;
    assume x in (X+p) (+) Y;
    then consider x2,y2 being Point of T such that
A1: x = x2+y2 and
A2: x2 in X+p and
A3: y2 in Y;
    consider x3 being Point of T such that
A4: x2 = x3+p & x3 in X by A2;
    x=x3+y2+p & x3+y2 in X (+) Y by A1,A3,A4,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
A5: x = x2+p and
A6: x2 in X (+) Y;
  consider x3,y3 being Point of T such that
A7: x2 = x3+y3 and
A8: x3 in X and
A9: y3 in Y by A6;
A10: x3+p in X+p by A8;
  x=x3+p+y3 by A5,A7,RLVECT_1:def 3;
  hence thesis by A9,A10;
end;
