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 SNoS_ : set set axiom SNoS_I2: !x:set.!y:set.SNo x -> SNo y -> SNoLev x iIn SNoLev y -> x iIn SNoS_ (SNoLev y) const SNoR : set set axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P axiom SNoR_SNoS: !x:set.SNo x -> !y:set.y iIn SNoR x -> y iIn SNoS_ (SNoLev x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x const SNoL : set set 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 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * x2) -> (!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL x2 -> (y2 * x2 + x * z2) < x * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR x2 -> (y2 * x2 + x * z2) < x * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR x2 -> (x * x2 + y2 * z2) < y2 * x2 + x * z2) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL x2 -> (x * x2 + y2 * z2) < y2 * x2 + x * z2) -> P) -> P) -> (!x2:set.x2 iIn SNoL x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> z iIn SNoL y -> SNoLev z iIn SNoLev y -> w iIn SNoR y -> SNo (x * z) -> SNo (x * w) -> u iIn SNoL x -> SNo (u * z) -> SNo (u * w) -> v iIn SNoL w -> v iIn SNoR z -> SNo v -> SNoLev v iIn SNoLev z -> v iIn SNoS_ (SNoLev y) -> (x * z + u * w) < u * z + x * w var x:set var y:set var z:set var w:set var u:set hyp SNo y hyp !v:set.v iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * v) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL v -> (x2 * v + x * y2) < x * v + x2 * y2) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoR v -> (x2 * v + x * y2) < x * v + x2 * y2) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR v -> (x * v + x2 * y2) < x2 * v + x * y2) -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL v -> (x * v + x2 * y2) < x2 * v + x * y2) -> P) -> P hyp !v:set.v iIn SNoL x -> !x2:set.SNo x2 -> SNo (v * x2) hyp z iIn SNoL y hyp SNo z hyp SNoLev z iIn SNoLev y hyp w iIn SNoR y hyp SNo w hyp SNo (x * z) hyp SNo (x * w) hyp z < w hyp u iIn SNoL x hyp SNo (u * z) claim SNo (u * w) -> (x * z + u * w) < u * z + x * w