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 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 exp_SNo_nat : set set set const ordsucc : set set const Empty : set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const omega : set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.SNo x -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) z) -> ordsucc Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) z * x -> SNo y -> w iIn omega -> y < w -> SNo w -> SNo (w * exp_SNo_nat (ordsucc (ordsucc Empty)) z) -> y <= (w * exp_SNo_nat (ordsucc (ordsucc Empty)) z) * x var x:set var y:set var z:set var w:set hyp SNo x hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) z) hyp ordsucc Empty <= exp_SNo_nat (ordsucc (ordsucc Empty)) z * x hyp SNo y hyp w iIn omega hyp y < w claim SNo w -> y <= (w * exp_SNo_nat (ordsucc (ordsucc Empty)) z) * x