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 Th17:
  for F being Ring, V, W being VectSp of F,
      T being linear-transformation of V,W
  for x,y being Element of V st T.x = T.y holds x - y in ker T
proof
  let F be Ring, V, W be VectSp of F,
      T be linear-transformation of V,W;
  let x,y be Element of V such that
A1: T.x = T.y;
  T.(x - y) = T.x - T.y by Th8
    .= 0.W by A1,VECTSP_1:19;
  hence thesis by Th10;
end;
