reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;

theorem Th19:
  for F being Ring, V, W being VectSp of F
  for X being Subset of V st V is Subspace of W holds X is Subset of W
proof
  let F be Ring, V, W be VectSp of F;
  let X be Subset of V;
  assume V is Subspace of W; then
A1: [#]V c= [#]W by VECTSP_4:def 2;
  X c= [#]W by A1;
  hence thesis;
end;
