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 Th9:
  for F being Ring,
      V, W being VectSp of F,
      T being linear-transformation of V,W holds
  T.(0.V) = 0.W
proof
  let F be Ring,
      V, W be VectSp of F,
      T be linear-transformation of V,W;
  0.V = (0.F)*(0.V) by VECTSP_1:14;
  then T.(0.V) = (0.F)*T.(0.V) by MOD_2:def 2
    .= 0.W by VECTSP_1:14;
  hence thesis;
end;
