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, m be Nat
  for M be Matrix of n,m,F_Real
  for A be linearly-independent Subset of REAL-NS m
    st the_rank_of M = n
  holds
    (Mx2Tran M) " A is linearly-independent
  proof
    let n, m be Nat;
    let M be Matrix of n,m,F_Real;
    let A be linearly-independent Subset of REAL-NS m;
    assume
    A1: the_rank_of M = n;
    reconsider B = A as linearly-independent Subset of TOP-REAL m
      by Th4,Th28;

    (Mx2Tran M) " B is linearly-independent by A1,MATRTOP2:26;
    hence
    (Mx2Tran M) " A is linearly-independent;
  end;
