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 A,B be linearly-independent Subset of REAL-NS n
    st card A = card B
  holds
    ex M be Matrix of n,F_Real
    st M is invertible
     & (Mx2Tran M) .: ([#] (Lin A)) = [#] (Lin B)
  proof
    let A,B be linearly-independent Subset of REAL-NS n;
    assume
    A1: card A = card B;

    reconsider A0 = A, B0 = B as Subset of TOP-REAL n by Th4;

    A0 is linearly-independent Subset of TOP-REAL n &
    B0 is linearly-independent Subset of TOP-REAL n by Th28; then
    consider M be Matrix of n,F_Real such that
    A2: M is invertible
     & (Mx2Tran M) .: ([#] (Lin A0)) = [#] (Lin B0) by A1,MATRTOP2:22;

    take M;
    thus M is invertible by A2;

    [#] (Lin A0) = [#] (Lin A) &
    [#] (Lin B0) = [#] (Lin B) by Th26;
    hence thesis by A2;
  end;
