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;

theorem Th23:
  (0).V is finite-dimensional
proof
  reconsider V9= (0).V as strict VectSp of GF;
  reconsider I = {}(the carrier of V9) as finite Subset of V9;
  the carrier of V9 = {0.V} by VECTSP_4:def 3
    .= {0.V9} by VECTSP_4:11
    .= the carrier of (0).V9 by VECTSP_4:def 3;
  then
A1: V9 = (0).V9 by VECTSP_4:31;
  Lin(I) = (0).V9 by VECTSP_7:9;
  then I is Basis of V9 by A1,VECTSP_7:def 3;
  hence thesis by MATRLIN:def 1;
end;
