Maximize :: $A = B \times C$ asks for two decimal numbers which, combined, use the digits from $1$ through $9$ exactly once, and which have the greatest possible product. (Actually, it asks for the product, but I want the factors).
kba's answer to that question suggests a procedure for finding these, but does not provide a rigorous proof that it works.
The maximum product of two numbers which together contain all non-zero digits exactly once is a duplicate with an answer that proves one aspect.
How can we generalize the problem/solution to arbitrary number bases and find a rigorous proof that it works? If possible, come up with further generalizations. For example: what happens if only certain digits may be used? A small bounty will be offered after two days.