Let $H$ be a complex Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$.
Set $H_0:=H_1\cap H_2\cap...\cap H_n$ and let
$P_i$ be the orthogonal projection onto $H_i$, $i=0,1,2,...,n$.
I study the functions $f_n:[0,1]\to\mathbb{R}$ defined by
$$
f_n(c)=\sup\{\|P_n...P_2 P_1-P_0\|\,|\,
c_F(H_1,...,H_n)\leqslant c\},\,c\in[0,1],
$$
where the supremum is taken over all complex Hilbert spaces $H$ and
systems of closed subspaces $H_1,...,H_n$ of $H$
for which the Friedrichs number $c_F(H_1,...,H_n)$ is less than or equal to $c$
(the Friedrichs number is a certain numerical characteristics of a system of subspaces).
Note that all such systems of subspaces **do not** form a set.
Despite of this, the function $f_n$ is well-defined
(see my Question Can we take a supremum over all Hilbert spaces?).
Indeed, let $A_{n}(c)$ be the set of all $a\in\mathbb{R}$ for which there exist a complex Hilbert space $H$ and a system of closed subspaces $H_1,...,H_n$ of $H$ such that
$c_F(H_1,...,H_n)\leqslant c$ and $\|P_n...P_2 P_1-P_0\|=a$.
Then by the axiom (scheme) of separation $A_{n}(c)$ is a **set** and thus we can take its supremum.

I need to show that $f_n(c)\leqslant g_n(c)$ for some function $g_n$. I argue as follows. Consider arbitrary element $a\in A_{n}(c)$. Then there exist a complex Hilbert space $H$ and a system of closed subspaces $H_1,...,H_n$ of $H$ such that $c_F(H_1,...,H_n)\leqslant c$ and $\|P_n...P_2 P_1-P_0\|=a$. After this I work with this system of subspaces $(H;H_1,...,H_n)$ and show that $\|P_n...P_2 P_1-P_0\|\leqslant g_n(c)$. Thus $a\leqslant g_n(c)$. Since this inequality holds for every $a\in A_{n}(c)$, we conclude that $\sup A_{n}(c)\leqslant g_n(c)$, i.e., $f_n(c)\leqslant g_n(c)$.

**Questions.** Are all these arguments correct, say, in the axiomatic theory ZFC?
I am suspicious here because we need to choose an element (system of subspaces)
from a class (and work with this element).
Do we need here something like the Axiom of Choice?

In response to Nate Eldredge's comment: essentially, the core of my worries is the following. Unfortunately, and this is the worst, is that I do not understand if the **function** $A_n(c)\ni a\mapsto (H;H_1,...,H_n)$
such that $c_F(H_1,...,H_n)\leqslant c$ and $\|P_n...P_2 P_1-P_0\|=a$,
is needed in the arguments above or not.

Please help me.