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 SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const SNoS_ : set set const omega : set const ap : set set set const SNoCut : set set set const Repl : set (set set) set const minus_SNo : set set term - = minus_SNo 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 -> (!w:set.w iIn omega -> SNo (ap y w)) -> (!w:set.w iIn omega -> SNo (ap z w)) -> (!w:set.w iIn omega -> !u:set.u iIn omega -> ap y w < ap z u) -> x iIn real var x:set var y:set var z:set hyp SNo x hyp y iIn setexp (SNoS_ omega) omega hyp z iIn setexp (SNoS_ omega) omega hyp !w:set.w iIn omega -> ap y w < x hyp !w:set.w iIn omega -> !u:set.u iIn w -> ap y u < ap y w hyp !w:set.w iIn omega -> x < ap z w hyp !w:set.w iIn omega -> !u:set.u iIn w -> ap z w < ap z u hyp x = SNoCut (Repl omega (ap y)) (Repl omega (ap z)) hyp SNo - x hyp !w:set.w iIn omega -> SNo (ap y w) claim (!w:set.w iIn omega -> SNo (ap z w)) -> x iIn real