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 axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const SNoL : set set axiom SNoL_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> y < x -> y iIn SNoL x 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 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.!x2:set.SNo x -> (!y2:set.y2 iIn SNoL x -> !z2:set.SNo z2 -> SNo (y2 * z2)) -> (!y2:set.y2 iIn SNoL x -> SNo (y2 * y)) -> SNo w -> SNo u -> SNoLev u iIn SNoLev x -> u < x -> SNo (z * y) -> SNo (x * w) -> SNo (z * w) -> SNo (u * y) -> SNo (x * v) -> SNo (u * v) -> SNo (u * w) -> (x * w + u * v) < u * w + x * v -> x2 iIn SNoL u -> (x2 * y + u * w) < u * y + x2 * w -> (z * y + x2 * w) < x2 * y + z * w -> x2 iIn SNoL x -> (z * y + x * w + u * v) < u * y + x * v + z * w const SNoR : set set const SNoS_ : set set var x:set var y:set var z:set var w:set var u:set var v:set var x2:set hyp SNo x hyp !y2:set.y2 iIn SNoS_ (SNoLev x) -> !z2:set.SNo z2 -> !P:prop.(SNo (y2 * z2) -> (!w2:set.w2 iIn SNoL y2 -> !u2:set.u2 iIn SNoL z2 -> (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2) -> (!w2:set.w2 iIn SNoR y2 -> !u2:set.u2 iIn SNoR z2 -> (w2 * z2 + y2 * u2) < y2 * z2 + w2 * u2) -> (!w2:set.w2 iIn SNoL y2 -> !u2:set.u2 iIn SNoR z2 -> (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2) -> (!w2:set.w2 iIn SNoR y2 -> !u2:set.u2 iIn SNoL z2 -> (y2 * z2 + w2 * u2) < w2 * z2 + y2 * u2) -> P) -> P hyp SNo y hyp !y2:set.y2 iIn SNoL x -> !z2:set.SNo z2 -> SNo (y2 * z2) hyp !y2:set.y2 iIn SNoL x -> SNo (y2 * y) hyp z iIn SNoL x hyp w iIn SNoL y hyp SNo w hyp SNo u hyp SNoLev u iIn SNoLev x hyp u < x hyp SNo (z * y) hyp SNo (x * w) hyp SNo (z * w) hyp SNo (u * y) hyp SNo (x * v) hyp SNo (u * v) hyp SNo (u * w) hyp (x * w + u * v) < u * w + x * v hyp x2 iIn SNoR z hyp x2 iIn SNoL u hyp (x2 * y + u * w) < u * y + x2 * w claim (z * y + x2 * w) < x2 * y + z * w -> (z * y + x * w + u * v) < u * y + x * v + z * w