deffunc H1( object ) -> object = upper_bound (PreNorms (modetrans ($1,X,Y,Z)));
A1:
for z being object st z in BoundedBilinearOperators (X,Y,Z) holds
H1(z) in REAL
by XREAL_0:def 1;
thus
ex f being Function of (BoundedBilinearOperators (X,Y,Z)),REAL st
for x being object st x in BoundedBilinearOperators (X,Y,Z) holds
f . x = H1(x)
from FUNCT_2:sch 2(A1); verum