const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const SNo : set prop const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const Empty : set axiom mul_SNo_zeroR: !x:set.SNo x -> x * Empty = Empty const ordsucc : set set axiom SNo_1: SNo (ordsucc Empty) axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_0R: !x:set.SNo x -> x + Empty = x const SNoL : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!y:set.SNo x -> (!z:set.z iIn SNoL x -> !w:set.w iIn SNoL (ordsucc Empty) -> (z * ordsucc Empty + x * w) < x * ordsucc Empty + z * w) -> Empty iIn SNoL (ordsucc Empty) -> y iIn SNoL x -> SNo y -> y * ordsucc Empty + x * Empty = y -> x * ordsucc Empty + y * Empty = x * ordsucc Empty -> y < x * ordsucc Empty const SNoS_ : set set const SNoLev : set set var x:set var y:set hyp SNo x hyp !z:set.z iIn SNoS_ (SNoLev x) -> z * ordsucc Empty = z hyp !z:set.z iIn SNoL x -> !w:set.w iIn SNoL (ordsucc Empty) -> (z * ordsucc Empty + x * w) < x * ordsucc Empty + z * w hyp Empty iIn SNoL (ordsucc Empty) hyp y iIn SNoL x hyp SNo y claim y * ordsucc Empty + x * Empty = y -> y < x * ordsucc Empty