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 Th16:
  (X+x)+y = X+(x+y)
proof
  thus X+x+y c= X+(x+y)
  proof
    let p be object;
    assume p in (X+x)+y;
    then consider x2 being Point of T such that
A1: p = x2+y and
A2: x2 in X+x;
    consider x3 being Point of T such that
A3: x2 = x3+x and
A4: x3 in X by A2;
    p=x3+(x+y) by A1,A3,RLVECT_1:def 3;
    hence thesis by A4;
  end;
  let p be object;
  assume p in X+(x+y);
  then consider x2 being Point of T such that
A5: p = x2+(x+y) & x2 in X;
  p = x2 + x + y & x2+x in X+x by A5,RLVECT_1:def 3;
  hence thesis;
end;
