
theorem
  for K being Ring, V being LeftMod of K, A being finite Subset of V
  holds
  A is linearly-independent
  iff
  ex b being FinSequence of V
  st b is one-to-one & rng b = A &
  for r being FinSequence of K,
  rb being FinSequence of V
  st len r = len b & len rb = len b &
  ( for i being Nat st i in dom rb holds rb.i = r/.i * b/.i) &
  Sum (rb) = 0.V
  holds r = Seg (len r) --> 0.K
  proof
    let K be Ring,
    V be LeftMod of K,
    A be finite Subset of V;
    hereby
      assume AS: A is linearly-independent;
      rng canFS (A) = A by FUNCT_2:def 3;
      then reconsider b = canFS (A) as FinSequence of the carrier of V
      by FINSEQ_1:def 4;
      take b;
      thus b is one-to-one;
      thus rng b = A by FUNCT_2:def 3;
      hence for r being FinSequence of K,
      rb being FinSequence of V
      st len r = len b & len rb = len b &
      ( for i being Nat st i in dom rb
      holds rb.i = r/.i * b/.i) & Sum (rb) = 0.V
      holds r = Seg (len r) --> 0.K by LMBASE2,AS;
    end;
    given b being FinSequence of V such that
A1: b is one-to-one & rng b = A &
    for r being FinSequence of K, rb being FinSequence of V
    st len r = len b & len rb = len b &
    ( for i being Nat st i in dom rb holds rb.i = r/.i * b/.i) &
    Sum (rb) = 0.V holds r = Seg (len r) --> 0.K;
    thus thesis by A1,LMBASE2;
  end;
