const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term nIn = \x:set.\y:set.~ x iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const ordinal : set prop const omega : set axiom omega_ordinal: ordinal omega const ap : set set set axiom ap_Pi: !x:set.!f:set set.!y:set.!z:set.y iIn Pi x f -> z iIn x -> ap y z iIn f z const SNoS_ : set set const SNoLev : set set const SNo : set prop const SNo_ : set set prop axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const real : set const minus_SNo : set set term - = minus_SNo const SNoL : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const eps_ : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set const SNoCutP : set set prop const SNoCut : set set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.~ x * y iIn real -> SNo x -> SNo y -> SNo (x * y) -> SNo - x * y -> (!x2:set.x2 iIn SNoL x -> !P:prop.(!y2:set.y2 iIn omega -> eps_ y2 <= x + - x2 -> P) -> P) -> (!x2:set.x2 iIn SNoR x -> !P:prop.(!y2:set.y2 iIn omega -> eps_ y2 <= x2 + - x -> P) -> P) -> (!x2:set.x2 iIn SNoL y -> !P:prop.(!y2:set.y2 iIn omega -> eps_ y2 <= y + - x2 -> P) -> P) -> (!x2:set.x2 iIn SNoR y -> !P:prop.(!y2:set.y2 iIn omega -> eps_ y2 <= x2 + - y -> P) -> P) -> SNoCutP z w -> (!x2:set.x2 iIn z -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn w -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P) -> x * y = SNoCut z w -> u iIn setexp (SNoS_ omega) omega -> (!x2:set.x2 iIn omega -> ap u x2 < x * y & x * y < ap u x2 + eps_ x2 & !y2:set.y2 iIn x2 -> ap u y2 < ap u x2) -> v iIn setexp (SNoS_ omega) omega -> (!x2:set.x2 iIn omega -> (ap v x2 + - eps_ x2) < x * y & x * y < ap v x2 & !y2:set.y2 iIn x2 -> ap v x2 < ap v y2) -> (!x2:set.x2 iIn omega -> SNo (ap u x2)) -> ~ !x2:set.x2 iIn omega -> SNo (ap v x2) var x:set var y:set var z:set var w:set var u:set var v:set hyp ~ x * y iIn real hyp SNo x hyp SNo y hyp SNo (x * y) hyp SNo - x * y hyp !x2:set.x2 iIn SNoL x -> !P:prop.(!y2:set.y2 iIn omega -> eps_ y2 <= x + - x2 -> P) -> P hyp !x2:set.x2 iIn SNoR x -> !P:prop.(!y2:set.y2 iIn omega -> eps_ y2 <= x2 + - x -> P) -> P hyp !x2:set.x2 iIn SNoL y -> !P:prop.(!y2:set.y2 iIn omega -> eps_ y2 <= y + - x2 -> P) -> P hyp !x2:set.x2 iIn SNoR y -> !P:prop.(!y2:set.y2 iIn omega -> eps_ y2 <= x2 + - y -> P) -> P hyp SNoCutP z w hyp !x2:set.x2 iIn z -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P hyp !x2:set.x2 iIn w -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P hyp x * y = SNoCut z w hyp u iIn setexp (SNoS_ omega) omega hyp !x2:set.x2 iIn omega -> ap u x2 < x * y & x * y < ap u x2 + eps_ x2 & !y2:set.y2 iIn x2 -> ap u y2 < ap u x2 hyp v iIn setexp (SNoS_ omega) omega hyp !x2:set.x2 iIn omega -> (ap v x2 + - eps_ x2) < x * y & x * y < ap v x2 & !y2:set.y2 iIn x2 -> ap v x2 < ap v y2 claim ~ !x2:set.x2 iIn omega -> SNo (ap u x2)