reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem
  for n be Nat,
      a be Element of REAL,
      a1 be Element of F_Real,
      x,y be Element of REAL n,
      x1,y1 be Element of n-tuples_on the carrier of F_Real
    st a = a1 & x = x1 & y = y1
  holds a * x = a1 * x1
      & x + y = x1 + y1;
