theorem Th20:
  f|X is bounded implies (BoundedFunctionsNorm X).f = upper_bound PreNorms f
proof
  reconsider f9=f as set;
  assume
A1: f|X is bounded;
  then f in BoundedFunctions X;
  then (BoundedFunctionsNorm X).f = upper_bound PreNorms(modetrans(f9,X))
   by Def17;
  hence thesis by A1,Th19;
end;
