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