reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th34:
  for X, Y be RealNormSpace for f being Point of
  R_NormSpace_of_BoundedLinearOperators(X,Y) st f = 0.
  R_NormSpace_of_BoundedLinearOperators(X,Y) holds 0 = ||.f.||
proof
  let X, Y be RealNormSpace;
  let f being Point of R_NormSpace_of_BoundedLinearOperators(X,Y) such that
A1: f = 0.R_NormSpace_of_BoundedLinearOperators(X,Y);
  thus ||.f.|| = 0
  proof
    reconsider g=f as Lipschitzian LinearOperator of X,Y by Def9;
    set z = (the carrier of X) --> 0.Y;
    reconsider z as Function of the carrier of X, the carrier of Y;
    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;
A4: PreNorms(g) is non empty bounded_above by Th27;
A5: z=g by A1,Th31;
A6: now
      let r be Real;
      assume r in PreNorms(g);
      then consider t be VECTOR of X such that
A7:   r=||.g.t.|| and
      ||.t.|| <= 1;
      ||.g.t.|| = ||.0.Y.|| by A5,FUNCOP_1:7
        .= 0;
      hence 0 <= r & r <=0 by A7;
    end;
    then 0 <= r0 by A2;
    then upper_bound PreNorms(g) = 0 by A6,A4,A2,A3,SEQ_4:def 1;
    then BoundedLinearOperatorsNorm(X,Y).f=0 by Th30;
    hence thesis;
  end;
end;
