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 add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) axiom add_SNo_Lt_subprop3b: !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo u -> SNo v -> (x + v) < z + u -> y < v + w -> (x + y) < z + w + u const SNoS_ : set set const SNoLev : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoL : set set const SNoR : set set 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)) -> (!x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> (!x2:set.x2 iIn SNoL y -> SNo (x * x2)) -> z iIn SNoL x -> w iIn SNoL y -> SNo w -> SNoLev w iIn SNoLev y -> w < y -> u iIn SNoR x -> v iIn SNoL y -> SNo v -> SNoLev v iIn SNoLev y -> v < y -> SNo (z * y) -> SNo (x * w) -> SNo (z * w) -> SNo (u * y) -> SNo (x * v) -> SNo (u * v) -> SNo (z * v) -> SNo (u * w) -> (z * y + u * w) < u * y + z * w -> (z * y + u * v) < u * y + z * v -> ((x * w + u * v) < u * w + x * v -> (z * y + x * w + u * v) < u * y + x * v + z * w) -> (z * y + x * w + u * v) < u * y + x * v + z * w var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo y hyp !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 hyp !x2:set.x2 iIn SNoL x -> !y2:set.SNo y2 -> SNo (x2 * y2) hyp !x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2) hyp !x2:set.x2 iIn SNoL y -> SNo (x * x2) hyp z iIn SNoL x hyp w iIn SNoL y hyp SNo w hyp SNoLev w iIn SNoLev y hyp w < y hyp u iIn SNoR x hyp v iIn SNoL y hyp SNo v hyp SNoLev v iIn SNoLev y hyp v < y 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 (z * v) hyp SNo (u * w) hyp !x2:set.x2 iIn SNoL y -> (z * y + u * x2) < u * y + z * x2 hyp (z * y + u * w) < u * y + z * w claim (z * y + u * v) < u * y + z * v -> (z * y + x * w + u * v) < u * y + x * v + z * w