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 Th19:
  for F be one-to-one FinSequence of TOP-REAL n st
    rng F is linearly-independent
  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 TOP-REAL n such that
   A1: rng F is linearly-independent;
  reconsider f=F as FinSequence of n-VectSp_over F_Real by Lm1;
  set M=FinS2MX f;
  lines M is linearly-independent by A1,Th7;
  then A2: the_rank_of M=len F by MATRIX13:121;
  then consider A be Matrix of n-' len F,n,F_Real such that
   A3: the_rank_of(M^A)=n by MATRTOP1:16;
  len F<=width M by A2,MATRIX13:74;
  then len F<=n by MATRIX_0:23;
  then n-len F=n-' len F by XREAL_1:233;
  then reconsider MA=M^A as Matrix of n,F_Real;
  take MA;
  Det MA<>0.F_Real by A3,MATRIX13:83;
  hence MA is invertible by LAPLACE:34;
  thus F=MA|dom F by FINSEQ_1:21
   .=MA|len F by FINSEQ_1:def 3;
end;
