reserve X,Y,Z for non trivial RealBanachSpace;

theorem
  for X,Y be RealNormSpace,
      v be Lipschitzian LinearOperator of X,Y,
      w be Point of R_NormSpace_of_BoundedLinearOperators (X,Y),
      a be Real
  st v = w
  holds -w = -v
  proof
    let X,Y be RealNormSpace,
        v be Lipschitzian LinearOperator of X,Y,
        w be Point of R_NormSpace_of_BoundedLinearOperators (X,Y),
        a be Real;
    assume
    A1: v = w;
    thus -w = (-1)*w by RLVECT_1:16
      .= (-1)(#)v by A1,XXXX
      .= -v by VFUNCT_1:23;
  end;
