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 Th32:
  for F be one-to-one FinSequence of REAL-NS n
    st rng F is linearly-independent
  holds
    ex M be Matrix of n,F_Real
    st M is invertible & M | len F = F
  proof
    let F be one-to-one FinSequence of REAL-NS n;
    assume
    A1: rng F is linearly-independent;
    reconsider F0 = F as FinSequence of TOP-REAL n by Th4;
    rng F0 is linearly-independent by A1,Th28; then
    consider M be Matrix of n,F_Real such that
    A2: M is invertible & M | len F0 = F0 by MATRTOP2:19;
    take M;
    thus thesis by A2;
  end;
