
theorem LMN9:
for X be RealNormSpace,
    a be Element of REAL, x be Point of DualSp X,
    v be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
  st x=v holds a*x = a*v
proof
   let X be RealNormSpace,
       a be Element of REAL, x be Point of DualSp X,
       v be Point of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real);
   assume AS: x=v;
   reconsider z=a*x as Point of DualSp X;
   reconsider u=a*v 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 = a*x.t by DUALSP01:30
       .= a*v.t by AS,BINOP_2:def 11;
    hence thesis by LOPBAN_1:36;
   end;
   hence thesis by A5,FUNCT_2:def 1,A2;
end;
