const SNo : set prop const ordsucc : set set const Empty : set axiom SNo_1: SNo (ordsucc Empty) const exp_SNo_nat : set set set axiom exp_SNo_nat_0: !x:set.SNo x -> exp_SNo_nat x Empty = ordsucc Empty const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const nat_p : set prop axiom exp_SNo_nat_S: !x:set.SNo x -> !y:set.nat_p y -> exp_SNo_nat x (ordsucc y) = x * exp_SNo_nat x y axiom nat_ind: !p:set prop.p Empty -> (!x:set.nat_p x -> p x -> p (ordsucc x)) -> !x:set.nat_p x -> p x claim !x:set.SNo x -> !y:set.nat_p y -> SNo (exp_SNo_nat x y)