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Example (Continued)

Take arbitrary x; we proceed by induction over n.

We have x⁰ = 1 = (prod_{1 <= i <= 0} x) and thus the induction base holds.

We take arbitrary n in N and assume

(1) xⁿ = (prod_{1 <= i <= n} x).

We have to prove

(2) xⁿ⁺¹ = (prod_{1 <= i <= n+1} x).

We know

xn+1	=	(definition exponentiation)
x * xn	=	(1)
x * (prod1 <= i <= n x)	=	(definition (prod ))
(prod1 <= i <= n+1 x)

which implies (2).