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 Th8:
  for F being Ring,
      V, W being VectSp of F,
      T being linear-transformation of V,W
  for x,y being Element of V holds T.x - T.y = T.(x - y)
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;
A1: T.((-1.F)*y) = (-1.F)*(T.y) by MOD_2:def 2;
  T.(x - y) = T.x + T.(-y) & -y = (-1.F)*y by VECTSP_1:14,def 20;
  hence thesis by A1,VECTSP_1:14;
end;
