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 Th10:
  for F being Ring,
      V, W being VectSp of F,
      T being linear-transformation of V,W
  for x being Element of V holds x in ker T iff 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 V;
  thus x in ker T implies T.x = 0.W
  proof
    assume x in ker T;
    then
A1: x in [#]ker T;
    [#]ker T = { u where u is Element of V : T.u = 0.W } by Def1;
    then ex u being Element of V st u = x & T.u = 0.W by A1;
    hence thesis;
  end;
  assume T.x = 0.W;
  then x in { u where u is Element of V : T.u = 0.W };
  then x in [#]ker T by Def1;
  hence thesis;
end;
