const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const Empty : set const ordsucc : set set axiom SNoLt_0_1: Empty < ordsucc Empty const SNo : set prop const exp_SNo_nat : set set set axiom exp_SNo_nat_0: !x:set.SNo x -> exp_SNo_nat x Empty = ordsucc Empty axiom SNo_0: SNo Empty const nat_p : set prop axiom SNo_exp_SNo_nat: !x:set.SNo x -> !y:set.nat_p y -> SNo (exp_SNo_nat x y) const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom pos_mul_SNo_Lt: !x:set.!y:set.!z:set.SNo x -> Empty < x -> SNo y -> SNo z -> y < z -> x * y < x * z axiom mul_SNo_zeroR: !x:set.SNo x -> x * Empty = Empty 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 -> Empty < x -> !y:set.nat_p y -> Empty < exp_SNo_nat x y