theorem
  f|X is bounded iff PreNorms f is bounded_above
proof
  now
    reconsider K = upper_bound PreNorms f as Real;
    assume
A1: PreNorms f is bounded_above;
    take K;
    now
      let t be object;
      assume t in X /\ dom f;
      then |.f.t.| in PreNorms f;
      hence |.f.t.| <= K by A1,SEQ_4:def 1;
    end;
    hence f|X is bounded by RFUNCT_1:73;
  end;
  hence thesis by Th17;
end;
