
theorem
  for X be non empty set for Y be RealNormSpace for g be Function of X,
  the carrier of Y holds g is bounded iff PreNorms(g) is bounded_above
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let g be Function of X,the carrier of Y;
  now
    reconsider K=upper_bound PreNorms(g) as Real;
    assume
A1: PreNorms(g) is bounded_above;
A2: 0 <= K
    proof
      consider r0 be object such that
A3:   r0 in PreNorms(g) by XBOOLE_0:def 1;
      reconsider r0 as Real by A3;
      now
        let r be Real;
        assume r in PreNorms(g);
        then ex t be Element of X st r=||.g.t.||;
        hence 0 <= r;
      end;
      then 0 <= r0 by A3;
      hence thesis by A1,A3,SEQ_4:def 1;
    end;
    take K;
    now
      let t be Element of X;
      ||.g.t.|| in PreNorms(g);
      hence ||.g.t.|| <= K by A1,SEQ_4:def 1;
    end;
    hence g is bounded by A2;
  end;
  hence thesis by Th11;
end;
