
theorem Th19:
  for X being non empty set,
      x being Element of X,
      f being Function of X,COMPLEX,
      F being Point of C_Normed_Algebra_of_BoundedFunctions(X) st
        f = F & f | X is bounded holds |.f.x.| <= ||.F.||
proof
  let X be non empty set,
      x be Element of X,
      f be Function of X,COMPLEX,
      F be Point of C_Normed_Algebra_of_BoundedFunctions(X);
  assume that
A1: f = F and
A2: f | X is bounded;
  reconsider r=|.f.x.| as ExtReal;
A3:r in PreNorms f;
  PreNorms f is non empty bounded_above by Th11,A2; then
  r <= upper_bound PreNorms(f) by A3,SEQ_4:def 1;
  hence |.f.x.| <= ||.F.|| by A1,A2,Th13;
end;
