
theorem LMN11:
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;
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;
   reconsider x1=x as Lipschitzian linear-Functional of X
     by DUALSP01:def 10;
   reconsider v1=v as Lipschitzian LinearOperator of X,RNS_Real
     by LOPBAN_1:def 9,BX;
   now let r be Real;
    assume AS2: r in PreNorms(v1);
    PreNorms(v1) = {||.v.t.|| where t is VECTOR of X : ||.t.|| <= 1 }
      by LOPBAN_1:def 12; then
    consider t be VECTOR of X such that
B1:   r = ||. v.t .|| & ||.t.|| <= 1 by AS2;
    |. x1.t .| <= ||.x.|| * ||.t.|| by DUALSP01:26; then
B2: ||. v.t .|| <= ||.x.|| * ||.t.|| by AS,EUCLID:def 2;
    ||.x.|| * ||.t.|| <= ||.x.|| * 1 by B1,XREAL_1:64;
    hence r <= ||.x.|| by B1,B2,XXREAL_0:2;
   end; then
   upper_bound PreNorms(v1) <= ||.x.|| by SEQ_4:45; then
A4: ||.v.|| <= ||.x.|| by BX,LOPBAN_1:30;
   now let r be Real;
    assume r in PreNorms(x1); then
    consider t be VECTOR of X such that
B1:   r = |. x.t .| & ||.t.|| <= 1;
    ||. v1.t .|| <= ||.v.|| * ||.t.|| by LOPBAN_1:32; then
B2: |. x.t .| <= ||.v.|| * ||.t.|| by AS,EUCLID:def 2;
    ||.v.|| * ||.t.|| <= ||.v.|| * 1 by B1,XREAL_1:64;
    hence r <= ||.v.|| by B1,B2,XXREAL_0:2;
   end; then
   upper_bound PreNorms(x1) <= ||.v.|| by SEQ_4:45; then
   ||.x.|| <= ||.v.|| by DUALSP01:24;
   hence thesis by A4,XXREAL_0:1;
end;
