Simplifying this proof/method

[Kalcor]

Question:

Drazil is playing a math game with Varda.

Let's define for positive integer x as a product of factorials of its
digits. For example, .

First, they choose a decimal number a consisting of n digits that
contains at least one digit larger than 1. This number may possibly
start with leading zeroes. Then they should find maximum positive
number x satisfying following two conditions:

1. x doesn't contain neither digit 0 nor digit 1.

2. = f(x) = f(a)

Help friends find such number.

Input The first line contains an integer n (1 ≤ n ≤ 15) — the number
of digits in a.

The second line contains n digits of a. There is at least one digit in
a that is larger than 1. Number a may possibly contain leading zeroes.

Output Output a maximum possible integer satisfying the conditions
above. There should be no zeroes and ones in this number decimal
representation.

Examples
input
4
1234
output
33222

input
3
555
output
555

Proof:

We can observe that our answer won't contain digits 4,6,8,9, because we can always transform digits 4,6,8,9 to more digits as in the conclusion, and it makes the number larger.

Then, how can we make sure that the result is the largest after this transformation?
We can prove the following lemma:

For any positive integer x, if it can be written as the form (2!)c2 * (3!)c3 * (5!)c5 * (7!)c7, there will be only one unique way.

Suppose that there exists two ways to write down x in this form, we can assume that the two ways are (2!)a2 * (3!)a3 * (5!)a5 * (7!)a7 and (2!)b2 * (3!)b3 * (5!)b5 * (7!)b7.

We find the largest i such that ai ≠ bi, Then we know there exists at least one prime number whose factor is different in the two ways.

But according to the Fundamental Theorem of Arithmetic, there is only one prime factorization of each integer. So we get a contradiction.

After getting the result, we don't need to worry about other numbers being larger than ours.

Time Complexity: O(n).

I'd like if someone explains the reason why were the digits transformed into other form of digits rather than just with some way to compute the number with..sadly brute force would give a TLE(Time limit exceeded) in this question cause of the 15 digit thing so that's a big number to brute force to,so I kindly hope that someone can explain the "proof" above, cause idk what theory says that these numbers can be transformed to those numbers for example 4 to 322 and stuff.
Thanks in advance.

[Kalcor]

Understood it, marked as solved!