reserve V for non empty RealLinearSpace;
reserve S for Real_Sequence;
reserve k,n,m,m1 for Nat;
reserve g,h,r,x for Real;

theorem Th30:
  for X be RealNormSpace, f be Lipschitzian linear-Functional of X
    holds (BoundedLinearFunctionalsNorm X).f = upper_bound PreNorms f
proof
  let X be RealNormSpace;
  let f be Lipschitzian linear-Functional of X;
  reconsider f9=f as set;
  f in BoundedLinearFunctionals X by Def9;
  hence (BoundedLinearFunctionalsNorm X).f =
  upper_bound PreNorms(Bound2Lipschitz(f9,X)) by Def13
    .= upper_bound PreNorms f by Th29;
end;
