reserve GF for Field,
  V for VectSp of GF,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n, m for Nat;
reserve V for finite-dimensional VectSp of GF,
  W, W1, W2 for Subspace of V,
  u, v for Vector of V;

theorem
  dim V < n implies n Subspaces_of V = {}
proof
  assume that
A1: dim V < n and
A2: n Subspaces_of V <> {};
  consider x being object such that
A3: x in n Subspaces_of V by A2,XBOOLE_0:def 1;
  ex W being strict Subspace of V st W = x & dim W = n by A3,Def2;
  hence contradiction by A1,Th25;
end;
