const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoS_ : set set const omega : set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const ap : set set set const SNoCut : set set set const Repl : set (set set) set const real : set lemma !x:set.!y:set.!z:set.SNo x -> y iIn setexp (SNoS_ omega) omega -> z iIn setexp (SNoS_ omega) omega -> (!w:set.w iIn omega -> ap y w < x) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap y u < ap y w) -> (!w:set.w iIn omega -> x < ap z w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap z w < ap z u) -> x = SNoCut (Repl omega (ap y)) (Repl omega (ap z)) -> SNo - x -> x iIn real const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set claim !x:set.SNo x -> !y:set.y iIn setexp (SNoS_ omega) omega -> !z:set.z iIn setexp (SNoS_ omega) omega -> (!w:set.w iIn omega -> ap y w < x) -> (!w:set.w iIn omega -> x < ap y w + eps_ w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap y u < ap y w) -> (!w:set.w iIn omega -> x < ap z w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap z w < ap z u) -> x = SNoCut (Repl omega (ap y)) (Repl omega (ap z)) -> x iIn real