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 Th28:
  dim W <= dim V
proof
  reconsider V9= V as RealLinearSpace;
  set I = the Basis of V9;
  reconsider I as finite Subset of V by Th23;
  set A = the Basis of W;
  reconsider A as Subset of W;
 A is linearly-independent by RLVECT_3:def 3;
  then reconsider A9= A as finite Subset of V by Th14,Th23;
  Lin(I) = the RLSStruct of V9 & A is linearly-independent Subset of V by Th14,
RLVECT_3:def 3;
  then
A1: card A9 <= card I by Th22;
  dim W = card A by Def2;
  hence thesis by A1,Def2;
end;
