reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;

theorem
  for V1 be VectSp of K for A be finite Subset of V1 st dim Lin(A) =
  card A holds A is linearly-independent
proof
  let V1 be VectSp of K;
  let A be finite Subset of V1 such that
A1: dim Lin(A) = card A;
  set L=Lin(A);
  A c= the carrier of L
  proof
    let x be object;
    assume x in A;
    then x in L by VECTSP_7:8;
    hence thesis;
  end;
  then reconsider A9=A as Subset of L;
  Lin(A9)=L by VECTSP_9:17;
  then consider B be Subset of L such that
A2: B c= A9 and
A3: B is linearly-independent and
A4: Lin(B) = L by VECTSP_7:18;
  reconsider B as finite Subset of L by A2;
  B is Basis of L by A3,A4,VECTSP_7:def 3;
  then reconsider L as finite-dimensional VectSp of K by MATRLIN:def 1;
  card A = dim L by A1
    .= card B by A3,A4,VECTSP_9:26;
  then A=B by A2,CARD_2:102;
  hence thesis by A3,VECTSP_9:11;
end;
