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Take arbitrary x; we proceed by induction over n.
We have x0 = 1 = (prod1 <= i <= 0 x) and thus the induction base holds.
We take arbitrary n in N and assume
(1) xn = (prod1 <= i <= n x).
We have to prove
(2) xn+1 = (prod1 <= i <= n+1 x).
xn+1 = (definition exponentiation) x * xn = (1) x * (prod1 <= i <= n x) = (definition (prod )) (prod1 <= i <= n+1 x)
which implies (2).