deffunc H_{1}( object ) -> object = upper_bound (PreNorms (modetrans ($1,X,Y)));

A1: for z being object st z in ComplexBoundedFunctions (X,Y) holds

H_{1}(z) in REAL
by XREAL_0:def 1;

ex f being Function of (ComplexBoundedFunctions (X,Y)),REAL st

for x being object st x in ComplexBoundedFunctions (X,Y) holds

f . x = H_{1}(x)
from FUNCT_2:sch 2(A1);

hence ex b_{1} being Function of (ComplexBoundedFunctions (X,Y)),REAL st

for x being object st x in ComplexBoundedFunctions (X,Y) holds

b_{1} . x = upper_bound (PreNorms (modetrans (x,X,Y)))
; :: thesis: verum

A1: for z being object st z in ComplexBoundedFunctions (X,Y) holds

H

ex f being Function of (ComplexBoundedFunctions (X,Y)),REAL st

for x being object st x in ComplexBoundedFunctions (X,Y) holds

f . x = H

hence ex b

for x being object st x in ComplexBoundedFunctions (X,Y) holds

b