
theorem
for X be RealNormSpace,
    x be Point of DualSp X,
    v be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
  st x=v holds -x = -v
proof
   let X be RealNormSpace,
       x be Point of DualSp X,
       v be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real);
   assume AS: x=v;
A1: (-1) is Element of REAL by XREAL_0:def 1;
   thus -x = (-1)*x by RLVECT_1:16
          .= (-1)*v by AS,A1,LMN9
          .= -v by RLVECT_1:16;
end;
