
theorem Th23:
  for F being Field for W1,W2 being finite-dimensional VectSp of F
  st (Omega).W1 = (Omega).W2 for m being Nat holds m Subspaces_of W1 = m
  Subspaces_of W2
proof
  let F be Field;
  let W1,W2 be finite-dimensional VectSp of F such that
A1: (Omega).W1 = (Omega).W2;
  let m be Nat;
  (Omega).W1 is Subspace of (Omega).W2 by A1,VECTSP_4:24;
  then W1 is Subspace of (Omega).W2 by Th5;
  then W1 is Subspace of W2 by Th3;
  hence m Subspaces_of W1 c= m Subspaces_of W2 by VECTSP_9:38;
  (Omega).W2 is Subspace of (Omega).W1 by A1,VECTSP_4:24;
  then W2 is Subspace of (Omega).W1 by Th5;
  then W2 is Subspace of W1 by Th3;
  hence thesis by VECTSP_9:38;
end;
