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
    ex M be Matrix of n,REAL
    st M is invertible & M | len F = F
  proof
    let F be one-to-one FinSequence of REAL-NS n;
    assume rng F is linearly-independent;
    then consider M be Matrix of n,F_Real such that
    A1: M is invertible & M | len F = F by Th32;
    reconsider N = MXF2MXR M as Matrix of n,REAL;
    take N;
    thus N is invertible by A1,Th33;
    thus N | len F = F by A1;
  end;
