
theorem Th21:
  for X being non empty set,
      F being Point of C_Normed_Algebra_of_BoundedFunctions(X) st
        F = 0.(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);
  set z = X --> 0;
  reconsider z = X --> 0c as Function of X,COMPLEX;
  F in ComplexBoundedFunctions X;
  then consider g being Function of X,COMPLEX such that
A1:g = F and
A2:g | X is bounded;
A3:( not PreNorms g is empty & PreNorms g is bounded_above ) by A2,Th11;
  consider r0 being object such that
A4:r0 in PreNorms g by XBOOLE_0:def 1;
  reconsider r0 as Real by A4;
A5:(for s be Real st s in PreNorms g holds s <= 0) implies upper_bound
  PreNorms g <= 0 by SEQ_4:45;
  assume
A6: F = 0.C_Normed_Algebra_of_BoundedFunctions(X);
A7:now
   let r be Real;
   assume r in PreNorms g;
   then consider t be Element of X such that
A8:r=|.(g.t).|;
   z=g by A6,A1,Th18; then
   |.(g.t).| = |.(0).|
           .= 0;
   hence 0 <= r & r <=0 by A8;
  end; then
  0<=r0 by A4; then
  upper_bound (PreNorms g) = 0 by A7,A3,A4,A5,SEQ_4:def 1;
  hence 0 = ||.F.|| by A1,A2,Th13;
end;
