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P(n)=n-1 such that n-1 is a natural number

Following the steps:

Base case: n=2, P(2)=1 which is a natural number.

We fix a natural number n and assume that P(n)=n-1 is true.

So P(n+1)=(n+1)-1=n. We choose n to be a natural number, therefore this is true.

Is this proof complete? It seems rather...light.