reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;
reserve V for finite-dimensional RealLinearSpace,
  W, W1, W2 for Subspace of V,
  u, v for VECTOR of V;

theorem Th29:
  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,RLVECT_3:15;
  then reconsider B = A as linearly-independent Subset of W by A1,Th15,
TARSKI:def 3;
  W = Lin B by Th20;
  then reconsider B as Basis of W by RLVECT_3:def 3;
  card B = dim W by Def2;
  hence thesis;
end;
