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
  V is_the_direct_sum_of W1, W2 implies dim V = dim W1 + dim W2
proof
  assume
A1: V is_the_direct_sum_of W1, W2;
  then
A2: the ModuleStr of V = W1 + W2 by VECTSP_5:def 4;
  W1 /\ W2 = (0).V by A1,VECTSP_5:def 4;
  then (Omega).(W1 /\ W2) = (0).V by VECTSP_4:def 4
    .= (0).(W1 /\ W2) by VECTSP_4:36;
  then dim(W1 /\ W2) = 0 by Th29;
  then dim W1 + dim W2 = dim(W1 + W2) + 0 by Th32
    .= dim (Omega).V by A2,VECTSP_4:def 4
    .= dim V by Th27;
  hence thesis;
end;
