Fastest way to prove that polynomials satisfy UFD property

[Meta Kunt]

­Greetings,

I need the following statement. Let S be a finite subset of monic linear polynomials over a field K[X]; for example S = { X. X+1, X+3} let
F be the set of all maps from S to NN0.
Let g be a map from F -> K[X] that maps an f in F to the product of s in S ( s^f(s) )
Then the map g is injective.

This corresponds to a specialisation of the UFD theorem, where you can replace K with an UFD ring and the set S with a finite subset of non-associated polynomials.
Also you get that the map is not only injective, but if x =/= y, then g(x) and g(y) are not associated.
Again I don't need the full general theorem. but I just need the special application of the theorem above.

Is something like that in the database already. If not, what would be the fastest way to prove this restriction of a statement.

Cheers,
metakunt

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