reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;
reserve v,v1,v2,u,w for Vector of n-VectSp_over K,
  t,t1,t2 for Element of n -tuples_on the carrier of K,

  L for Linear_Combination of n-VectSp_over K,
  M,M1 for Matrix of m,n,K;

theorem
  for V be finite Subset of n-VectSp_over K ex M be Matrix of card V,n,K
  st M is without_repeated_line & lines M = V
proof
  let V be finite Subset of n-VectSp_over K;
  set cV=card V;
  card Seg cV=cV by FINSEQ_1:57;
  then Seg cV,V are_equipotent by CARD_1:5;
  then consider m be Function such that
A1: m is one-to-one and
A2: dom m = Seg cV and
A3: rng m=V by WELLORD2:def 4;
  reconsider M=m as FinSequence by A2,FINSEQ_1:def 2;
  now
    let x;
    assume x in rng M;
    then reconsider p=x as Element of n-tuples_on the carrier of K by A3,Th102;
    len p=n by CARD_1:def 7;
    hence ex p st x = p & len p = n;
  end;
  then reconsider M as Matrix of K by MATRIX_0:9;
A4: len M=cV by A2,FINSEQ_1:def 3;
  the carrier of n -VectSp_over K = n-tuples_on the carrier of K by Th102;
  then for p st p in rng M holds len p=n by A3,CARD_1:def 7;
  then M is Matrix of cV,n,K by A4,MATRIX_0:def 2;
  hence thesis by A1,A3;
end;
