reserve GF for Field,
  V for VectSp of GF,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n, m for Nat;
reserve V for finite-dimensional VectSp of GF,
  W, W1, W2 for Subspace of V,
  u, v for Vector of V;

theorem Th26:
  for A being Subset of V st A is linearly-independent holds
    card A = dim Lin(A)
proof
  let A be Subset of V such that
A1: A is linearly-independent;
  set W = Lin(A);
  for x being object st x in A holds x in the carrier of W
     by STRUCT_0:def 5,VECTSP_7:8;
  then reconsider B = A as linearly-independent Subset of W by A1,Th12,
TARSKI:def 3;
  W = Lin(B) by Th17;
  then reconsider B as Basis of W by VECTSP_7:def 3;
  card B = dim W by Def1;
  hence thesis;
end;
