
theorem
for X be RealNormSpace,
    x,y be Point of DualSp X,
    v,w be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
  st x=v & y=w holds x + y = v + w
proof
   let X be RealNormSpace,
       x,y be Point of DualSp X,
       v,w be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real);
   assume AS: x=v & y=w;
   reconsider z=x+y as Point of DualSp X;
   reconsider u=v+w as
     Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real);
BX:R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
     = NORMSTR (# BoundedLinearOperators(X,RNS_Real),
         Zero_(BoundedLinearOperators(X,RNS_Real),
             R_VectorSpace_of_LinearOperators(X,RNS_Real)),
         Add_(BoundedLinearOperators(X,RNS_Real),
             R_VectorSpace_of_LinearOperators(X,RNS_Real)),
         Mult_(BoundedLinearOperators(X,RNS_Real),
             R_VectorSpace_of_LinearOperators(X,RNS_Real)),
         BoundedLinearOperatorsNorm(X,RNS_Real) #) by LOPBAN_1:def 14;
A1:z is Lipschitzian additive homogeneous
        Function of the carrier of X,REAL by DUALSP01:def 10; then
A5:dom z = the carrier of X by FUNCT_2:def 1;
A2:u is Lipschitzian additive homogeneous Function of X,RNS_Real
      by LOPBAN_1:def 9,BX;
   for t be object st t in dom z holds z.t = u.t
   proof
    let t be object;
    assume t in dom z; then
    reconsider t as VECTOR of X by FUNCT_2:def 1,A1;
    z.t = x.t + y.t by DUALSP01:29
       .= v.t + w.t by AS,BINOP_2:def 9;
    hence thesis by LOPBAN_1:35;
   end;
   hence thesis by A2,A5,FUNCT_2:def 1;
end;
