const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const omega : set const ordsucc : set set axiom omega_ordsucc: !x:set.x iIn omega -> ordsucc x iIn omega const nat_p : set prop axiom omega_nat_p: !x:set.x iIn omega -> nat_p x axiom nat_ordsucc_in_ordsucc: !x:set.nat_p x -> !y:set.y iIn x -> ordsucc y iIn ordsucc x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_Lt3: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> x < z -> y < w -> (x + y) < z + w const ap : set set set const Sigma : set (set set) set var x:set var y:set var z:set var w:set hyp !u:set.u iIn omega -> !v:set.v iIn u -> ap x u < ap x v hyp !u:set.u iIn omega -> !v:set.v iIn u -> ap y u < ap y v hyp !u:set.u iIn omega -> SNo (ap x (ordsucc u)) hyp !u:set.u iIn omega -> SNo (ap y (ordsucc u)) hyp !u:set.u iIn omega -> ap (Sigma omega \v:set.ap x (ordsucc v) + ap y (ordsucc v)) u = ap x (ordsucc u) + ap y (ordsucc u) hyp z iIn omega hyp w iIn z claim w iIn omega -> (ap x (ordsucc z) + ap y (ordsucc z)) < ap (Sigma omega \u:set.ap x (ordsucc u) + ap y (ordsucc u)) w