
theorem Th15:
  for X be non empty set, Y be ComplexNormSpace, f be bounded
  Function of X,the carrier of Y holds ComplexBoundedFunctionsNorm(X,Y).f
   = upper_bound
  PreNorms(f)
proof
  let X be non empty set;
  let Y be ComplexNormSpace;
  let f be bounded Function of X,the carrier of Y;
  reconsider f9=f as set;
  f in ComplexBoundedFunctions(X,Y) by Def5;
  hence
  ComplexBoundedFunctionsNorm(X,Y).f = upper_bound PreNorms(modetrans(f9,X,Y))
   by Def9
    .= upper_bound PreNorms(f) by Th14;
end;
