theorem
  addMagma (# REAL, addreal #) is Abelian addGroup
proof
  reconsider G = addMagma (# REAL, addreal #) as addGroup by Th3;
  G is Abelian
  proof
    let h,g be Element of G;
    reconsider A = h, B = g as Real;
    thus h + g = B + A by BINOP_2:def 9
      .= g + h by BINOP_2:def 9;
  end;
  hence thesis;
end;
