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 Th25:
  dim W <= dim V
proof
  set A = the Basis of W;
  reconsider A as Subset of W;
A1: dim W = card A by Def1;
  A is linearly-independent by VECTSP_7:def 3;
  then reconsider B = A as linearly-independent Subset of V by Th11;
  reconsider A9= B as finite Subset of V;
  reconsider V9= V as VectSp of GF;
  set I = the Basis of V9;
A2: Lin(I) = the ModuleStr of V9 by VECTSP_7:def 3;
  reconsider I as finite Subset of V;
  card A9 <= card I by A2,Th19;
  hence thesis by A1,Def1;
end;
