
theorem LOPBAN73:
  for X, Y being RealLinearSpace
  for f being LinearOperator of X,Y
  holds 0.Y = f.(0. X)
  proof
    let X, Y be RealLinearSpace;
    let f be LinearOperator of X,Y;
    (f /. (0. X)) + (0. Y)
     = f /. ((0. X) + (0. X))
    .= (f /. (0. X)) + (f /. (0. X)) by VECTSP_1:def 20;
    hence 0.Y = f.(0. X) by RLVECT_1:8;
  end;
