theorem Th17:
  f|X is bounded implies PreNorms f is bounded_above
proof
  assume f|X is bounded;
  then consider K be Real such that
A1: for x be object st x in X /\ dom f holds |.f.x.| <= K by RFUNCT_1:73;
A2: X /\ dom f = X /\ X by FUNCT_2:def 1;
  take K;
    let r be ExtReal;
    assume r in PreNorms f;
    then ex t be Element of X st r=|.f.t.|;
    hence r <=K by A1,A2;
end;
