
theorem
  for X being non empty set,
      F being Point of C_Normed_Algebra_of_BoundedFunctions(X) holds
            0 <= ||.F.||
proof
  let X be non empty set,
      F be Point of C_Normed_Algebra_of_BoundedFunctions(X);
  F in ComplexBoundedFunctions X;
  then consider g being Function of X,COMPLEX such that
A1: F = g and
A2: g | X is bounded;
  consider r0 being object such that
A3: r0 in PreNorms g by XBOOLE_0:def 1;
  reconsider r1 = r0 as Real by A3;
  now
   let r be Real;
   assume r in PreNorms g;
   then ex t being Element of X st r = |.(g .t).|;
   hence 0 <= r;
  end;
  then
A4:  0 <= r1 by A3;
  PreNorms g is non empty bounded_above by Th11,A2;
  then 0 <= upper_bound (PreNorms g) by A3,A4,SEQ_4:def 1;
  hence 0 <= ||.F.|| by A1,A2,Th13;
end;
