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 Th33:
  for X, Y be RealNormSpace for f being Point of
  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);
  reconsider g=f as Lipschitzian LinearOperator of X,Y by Def9;
  consider r0 be object such that
A1: r0 in PreNorms(g) by XBOOLE_0:def 1;
  reconsider r0 as Real by A1;
A2: PreNorms(g) is non empty bounded_above by Th27;
A3: BoundedLinearOperatorsNorm(X,Y).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;
  end;
  then 0 <= r0 by A1;
  then 0 <=upper_bound PreNorms(g) by A2,A1,SEQ_4:def 1;
  hence thesis by A3;
end;
