deffunc H_{1}( object ) -> object = upper_bound (rng |.(seq_id $1).|);

A1: for z being object st z in the_set_of_BoundedComplexSequences holds

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

ex f being Function of the_set_of_BoundedComplexSequences,REAL st

for x being object st x in the_set_of_BoundedComplexSequences holds

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

hence ex b_{1} being Function of the_set_of_BoundedComplexSequences,REAL st

for x being object st x in the_set_of_BoundedComplexSequences holds

b_{1} . x = upper_bound (rng |.(seq_id x).|)
; :: thesis: verum

A1: for z being object st z in the_set_of_BoundedComplexSequences holds

H

ex f being Function of the_set_of_BoundedComplexSequences,REAL st

for x being object st x in the_set_of_BoundedComplexSequences holds

f . x = H

hence ex b

for x being object st x in the_set_of_BoundedComplexSequences holds

b