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
  for F being Ring, V, W being VectSp of F,
      T being linear-transformation of V,W
  for x being Element of ker T holds T.x = 0.W
proof
  let F be Ring, V, W be VectSp of F,
      T be linear-transformation of V,W;
  let x be Element of ker T;
  reconsider y = x as Element of V by VECTSP_4:10;
  y in ker T;
  hence thesis by Th10;
end;
