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 Th33:
  for X be RealNormSpace, f be Point of DualSp X holds 0 <= ||.f.||
proof
  let X be RealNormSpace;
  let f be Point of DualSp X;
  reconsider g=f as Lipschitzian linear-Functional of X by Def9;
  consider r0 be object such that
A1: r0 in PreNorms g by XBOOLE_0:def 1;
  reconsider r0 as Real by A1;
A3: (BoundedLinearFunctionalsNorm X).f = upper_bound PreNorms g by Th30;
  now
    let r be Real;
    assume r in PreNorms g;
    then ex t be VECTOR of X st r=|.g.t.| & ||.t.|| <= 1;
    hence 0 <= r by COMPLEX1:46;
  end;
  then 0 <= r0 by A1;
  hence thesis by A1,SEQ_4:def 1,A3;
end;
