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Proposition: In order to prove F, it suffices to prove
(forall y in N: y=T => F)where y does not occur freely in T or F.
Consequence: in order to prove
forall x0, ..., xn-1: Fwe may prove
(forall x0, ..., xn-1,y in N: y=T => F)where T is a term with free variables x0, ..., xn-1.
We introduce a variable over N to proceed by induction.