
theorem Th14:
  for X be non empty set for Y be RealNormSpace for f be bounded
Function of X,the carrier of Y holds BoundedFunctionsNorm(X,Y).f =
 upper_bound PreNorms
  (f)
proof
  let X be non empty set;
  let Y be RealNormSpace;
  let f be bounded Function of X,the carrier of Y;
  reconsider f9=f as set;
  f in BoundedFunctions(X,Y) by Def5;
  hence BoundedFunctionsNorm(X,Y).f = upper_bound PreNorms(modetrans(f9,X,Y))
  by Def9
    .= upper_bound PreNorms(f) by Th13;
end;
