
theorem
  for V being finite-dimensional RealUnitarySpace, W being Subspace of V
  , n being Element of NAT holds n Subspaces_of W c= n Subspaces_of V
proof
  let V be finite-dimensional RealUnitarySpace;
  let W be Subspace of V;
  let n be Element of NAT;
  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 RUSUB_1:21;
  W1 in n Subspaces_of V by A2,Def3;
  hence thesis by A1;
end;
