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
  n Subspaces_of W c= n Subspaces_of V
proof
  let x be object;
  assume x in n Subspaces_of W;
  then consider W1 being strict Subspace of W such that
A1: W1 = x and
A2: dim W1 = n by Def3;
  reconsider W1 as strict Subspace of V by RLSUB_1:27;
  W1 in n Subspaces_of V by A2,Def3;
  hence thesis by A1;
end;
