deffunc H₁( object ) -> object = upper_bound (rng |.(seq_id $1).|);
A1: for z being object st z in the_set_of_BoundedComplexSequences holds
H₁(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 = H₁(x) from FUNCT_2:sch 2(A1);
hence ex b₁ being Function of the_set_of_BoundedComplexSequences,REAL st
for x being object st x in the_set_of_BoundedComplexSequences holds
b₁ . x = upper_bound (rng |.(seq_id x).|) ; :: thesis: verum