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 one-to-one FinSequence of REAL-NS n
    st rng F is linearly-independent
  holds
    for B be OrdBasis of n -VectSp_over F_Real
      st B = MX2FinS(1. (F_Real,n))
    holds
      for M be Matrix of n,F_Real
        st M is invertible & M | len F = F
      holds
        (Mx2Tran M) .: ([#] (Lin (rng (B | len F))))
          = [#] (Lin rng F)
  proof
    let F be one-to-one FinSequence of REAL-NS n;
    assume
    A1: rng F is linearly-independent;
    let B be OrdBasis of n -VectSp_over F_Real;
    assume
    A2: B = MX2FinS(1. (F_Real,n));
    let M be Matrix of n,F_Real;
    assume
    A3: M is invertible & M | len F = F;
    reconsider F0 = F as FinSequence of TOP-REAL n by Th4;
    rng F0 is linearly-independent by A1,Th28; then
    (Mx2Tran M) .: ([#] (Lin (rng (B | len F0))))
      = [#] (Lin rng F0) by A2,A3,MATRTOP2:21;
    hence thesis by Th26;
  end;
