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 Af be Subset of n-VectSp_over F_Real,
      Ar be Subset of REAL-NS n
    st Af = Ar
  holds
    Af is linearly-independent
      iff
    Ar is linearly-independent
  proof
    let Af be Subset of n-VectSp_over F_Real,
        Ar be Subset of REAL-NS n;
    assume
    A1: Af = Ar;
    reconsider At = Ar as Subset of TOP-REAL n by Th4;
    Ar is linearly-independent
      iff
    At is linearly-independent by Th28;
    hence thesis by A1,MATRTOP2:7;
  end;
