
theorem Th13:
  for X being non empty set,
      f being Function of X,COMPLEX st f|X is bounded holds
  (ComplexBoundedFunctionsNorm(X)).f = upper_bound PreNorms f
proof
  let X be non empty set,
      f be Function of X,COMPLEX;
  assume
A1:f|X is bounded; then
  f in ComplexBoundedFunctions(X); then
  (ComplexBoundedFunctionsNorm(X)).f = upper_bound PreNorms(modetrans(f,X))
    by Def9;
  hence thesis by Th12,A1;
end;
