deffunc H_{1}( object ) -> object = upper_bound (PreNorms (Bound2Lipschitz ($1,X)));

A1: for z being object st z in BoundedLinearFunctionals X holds

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

thus ex f being Function of (BoundedLinearFunctionals X),REAL st

for x being object st x in BoundedLinearFunctionals X holds

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

A1: for z being object st z in BoundedLinearFunctionals X holds

H

thus ex f being Function of (BoundedLinearFunctionals X),REAL st

for x being object st x in BoundedLinearFunctionals X holds

f . x = H