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

theorem
  for V be VectSp of K for A be finite Subset of V holds dim Lin A <= card A
proof
  let V be VectSp of K;
  let A be finite Subset of V;
  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
A1: B c= A9 and
A2: B is linearly-independent & Lin(B) = L by VECTSP_7:18;
  reconsider B as finite Subset of L by A1;
  B is Basis of L by A2,VECTSP_7:def 3;
  then reconsider L as finite-dimensional VectSp of K by MATRLIN:def 1;
  card B = dim L & Segm card B c= Segm card A by A1,A2,CARD_1:11,VECTSP_9:26;
  hence thesis by NAT_1:39;
end;
