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 SNoL : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 var x:set var y:set hyp SNo x 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 hyp y * ordsucc Empty + x * Empty = y claim x * ordsucc Empty + y * Empty = x * ordsucc Empty -> y < x * ordsucc Empty