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 F be FinSequence of REAL-NS n,
      fr be Function of REAL-NS n,REAL,
      Fv be FinSequence of n-VectSp_over F_Real,
      fv be Function of n-VectSp_over F_Real,F_Real
    st fr = fv & F = Fv
  holds fr(#)F = fv(#)Fv
  proof
    let F be FinSequence of REAL-NS n,
       fr be Function of REAL-NS n,REAL,
       Fv be FinSequence of n-VectSp_over F_Real,
       fv be Function of n-VectSp_over F_Real,F_Real;
    assume
    A1: fr = fv & F = Fv;
    reconsider fr1 = fr as Function of TOP-REAL n,REAL by Th18;
    reconsider F1 = F as FinSequence of TOP-REAL n by Th4;
    fr1(#)F1 = fr(#)F by Th19;
    hence thesis by A1,MATRTOP2:3;
  end;
