
theorem
  for X be non empty set for Y be RealNormSpace for f being Point of
  R_NormSpace_of_BoundedFunctions(X,Y) holds 0 <= ||.f.||
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let f being Point of R_NormSpace_of_BoundedFunctions(X,Y);
  reconsider g=f as bounded Function of X,the carrier of Y by Def5;
  consider r0 be object such that
A1: r0 in PreNorms(g) by XBOOLE_0:def 1;
  reconsider r0 as Real by A1;
A2: PreNorms(g) is non empty bounded_above by Th11;
  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 A1;
  then 0 <=upper_bound PreNorms(g) by A2,A1,SEQ_4:def 1;
  hence thesis by Th14;
end;
