const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop const ordsucc : set set const Empty : set axiom SNo_1: SNo (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 SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 axiom nonneg_mul_SNo_Le: !x:set.!y:set.!z:set.SNo x -> Empty <= x -> SNo y -> SNo z -> y <= z -> x * y <= x * z const omega : set const exp_SNo_nat : set set set var x:set var y:set var z:set hyp SNo x hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) y) hyp ordsucc Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) y * x hyp z iIn omega hyp SNo z claim Empty <= z -> z * ordsucc Empty <= z * exp_SNo_nat (ordsucc (ordsucc Empty)) y * x