
theorem Th11:
  for X being non empty set,
      f being Function of X,COMPLEX holds
  f|X is bounded iff PreNorms f is bounded_above
proof
  let X be non empty set,
      f be Function of X,COMPLEX;
  now assume
A1:PreNorms f is bounded_above;
   reconsider K = upper_bound PreNorms f as Real;
A2:now let t be Element of X;
    assume t in X /\ dom f;
    |.f.t.| in PreNorms f;
    hence |.(f/.t).| <= K by A1,SEQ_4:def 1;
   end;
   take K;
   thus f|X is bounded by A2,CFUNCT_1:69;
  end;
  hence thesis by Th10;
end;
