const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const ordinal : set prop const SNoLev : set set axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x) axiom ordinal_TransSet: !x:set.ordinal x -> TransSet x const SNoL : set set axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const SNoR : set set const SNoS_ : set set axiom SNoR_SNoS: !x:set.SNo x -> !y:set.y iIn SNoR x -> y iIn SNoS_ (SNoLev x) axiom SNoL_SNoS: !x:set.SNo x -> !y:set.y iIn SNoL x -> y iIn SNoS_ (SNoLev x) axiom SNoLt_SNoL_or_SNoR_impred: !x:set.!y:set.SNo x -> SNo y -> x < y -> !P:prop.(!z:set.z iIn SNoL y -> z iIn SNoR x -> P) -> (x iIn SNoL y -> P) -> (y iIn SNoR x -> P) -> P const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.SNo y2 -> !P:prop.(SNo (x2 * y2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoL y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoR y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoR y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoL y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> P) -> P) -> SNo y -> z iIn SNoR x -> SNoLev z iIn SNoLev x -> w iIn SNoL x -> SNo (z * y) -> SNo (w * y) -> u iIn SNoR y -> SNo u -> SNo (z * u) -> SNo (w * u) -> v iIn SNoL z -> v iIn SNoR w -> SNo v -> SNoLev v iIn SNoLev z -> SNoLev v iIn SNoLev x -> (z * y + w * u) < w * y + z * u var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp !v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.SNo x2 -> !P:prop.(SNo (v * x2) -> (!y2:set.y2 iIn SNoL v -> !z2:set.z2 iIn SNoL x2 -> (y2 * x2 + v * z2) < v * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoR v -> !z2:set.z2 iIn SNoR x2 -> (y2 * x2 + v * z2) < v * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoL v -> !z2:set.z2 iIn SNoR x2 -> (v * x2 + y2 * z2) < y2 * x2 + v * z2) -> (!y2:set.y2 iIn SNoR v -> !z2:set.z2 iIn SNoL x2 -> (v * x2 + y2 * z2) < y2 * x2 + v * z2) -> P) -> P hyp SNo y hyp !v:set.v iIn SNoL x -> !x2:set.SNo x2 -> SNo (v * x2) hyp z iIn SNoR x hyp SNo z hyp SNoLev z iIn SNoLev x hyp w iIn SNoL x hyp SNo w hyp SNo (z * y) hyp SNo (w * y) hyp w < z hyp u iIn SNoR y hyp SNo u hyp SNo (z * u) claim SNo (w * u) -> (z * y + w * u) < w * y + z * u