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 term nIn = \x:set.\y:set.~ x iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x 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 omega : set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const eps_ : set set const ordsucc : set set const Empty : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> z iIn omega -> eps_ z * x < ordsucc Empty -> eps_ z * y < ordsucc Empty -> w iIn omega -> w + ordsucc Empty iIn omega -> w + ordsucc (ordsucc Empty) iIn omega -> u < x -> SNo u -> v < y -> SNo v -> SNo (eps_ z) -> SNo (eps_ (w + ordsucc Empty)) -> SNo (eps_ (w + ordsucc (ordsucc Empty))) -> SNo (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) -> SNo ((eps_ z * eps_ (w + ordsucc (ordsucc Empty))) * eps_ z * eps_ (w + ordsucc (ordsucc Empty))) -> (u * eps_ z * eps_ (w + ordsucc (ordsucc Empty)) + (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) * v + (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) * eps_ z * eps_ (w + ordsucc (ordsucc Empty))) < (eps_ (w + ordsucc (ordsucc Empty)) + eps_ (w + ordsucc (ordsucc Empty))) + eps_ (w + ordsucc Empty) var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp SNo y hyp z iIn omega hyp eps_ z * x < ordsucc Empty hyp eps_ z * y < ordsucc Empty hyp w iIn omega hyp w + ordsucc Empty iIn omega hyp w + ordsucc (ordsucc Empty) iIn omega hyp u < x hyp SNo u hyp v < y hyp SNo v hyp SNo (eps_ z) hyp SNo (eps_ (w + ordsucc Empty)) hyp SNo (eps_ (w + ordsucc (ordsucc Empty))) claim SNo (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) -> (u * eps_ z * eps_ (w + ordsucc (ordsucc Empty)) + (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) * v + (eps_ z * eps_ (w + ordsucc (ordsucc Empty))) * eps_ z * eps_ (w + ordsucc (ordsucc Empty))) < (eps_ (w + ordsucc (ordsucc Empty)) + eps_ (w + ordsucc (ordsucc Empty))) + eps_ (w + ordsucc Empty)