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;

theorem Th25:
  V is finite-dimensional implies for A, B being Basis of V holds
  card A = card B
proof
  assume
A1: V is finite-dimensional;
  let A, B be Basis of V;
  reconsider A9= A, B9= B as finite Subset of V by A1,Th23;
  the RLSStruct of V = Lin(B) & A9 is linearly-independent by RLVECT_3:def 3;
  then
A2: card A9 <= card B9 by Th22;
  the RLSStruct of V = Lin(A) & B9 is linearly-independent by RLVECT_3:def 3;
  then card B9 <= card A9 by Th22;
  hence thesis by A2,XXREAL_0:1;
end;
