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 (+) Z) = (X (+) Y) (+) Z
proof
  thus X (+) ( Y (+) Z) c= (X (+) Y) (+) Z
  proof
    let p be object;
    assume p in X (+) ( Y (+) Z);
    then consider x1,p2 being Point of T such that
A1: p = x1+p2 & x1 in X and
A2: p2 in Y (+) Z;
    consider y,z being Point of T such that
A3: p2=y+z & y in Y and
A4: z in Z by A2;
    set p3=x1+y;
    p=x1+y+z & p3 in X (+) Y by A1,A3,RLVECT_1:def 3;
    hence thesis by A4;
  end;
  let p be object;
  assume p in (X (+) Y) (+) Z;
  then consider x1,p2 being Point of T such that
A5: p = x1+p2 and
A6: x1 in X (+) Y and
A7: p2 in Z;
  consider y,z being Point of T such that
A8: x1=y+z and
A9: y in X and
A10: z in Y by A6;
  set p3=z+p2;
  p=y+(z+p2) & p3 in Y (+) Z by A5,A7,A8,A10,RLVECT_1:def 3;
  hence thesis by A9;
end;
