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 Th34:
  for X, Y be RealNormSpace for f be Point of DualSp X
    st f = 0.(DualSp X) holds 0 = ||.f.||
proof
  let X, Y be RealNormSpace;
  let f be Point of DualSp X such that
A1: f = 0.(DualSp X);
  thus ||.f.|| = 0
  proof
    reconsider g=f as Lipschitzian linear-Functional of X by Def9;
    set z = (the carrier of X) --> 0;
    reconsider z as Function of the carrier of X,REAL
    by FUNCOP_1:45,XREAL_0:def 1;
    consider r0 be object such that
A2: r0 in PreNorms g by XBOOLE_0:def 1;
    reconsider r0 as Real by A2;
A3: (for s be Real st s in PreNorms g holds s <= 0) implies upper_bound
    PreNorms g <= 0 by SEQ_4:45;
A5: z=g by A1,Th31;
A6: 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 & r <=0 by A5,COMPLEX1:44;
    end;
    then 0 <= r0 by A2;
    then upper_bound PreNorms g = 0 by A6,A2,A3,SEQ_4:def 1;
    hence thesis by Th30;
  end;
end;
