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 Th12:
  for F being Ring, V, W being VectSp of F,
      T being linear-transformation of V,W holds
  for X being Subset of V holds T .: X is Subset of im T
proof
  let F be Ring, V, W be VectSp of F,
      T be linear-transformation of V,W;
  let X be Subset of V;
  [#](im T) = T .: [#]V by Def2;
  hence thesis by RELAT_1:123;
end;
