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_subprop3c: !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.SNo x -> SNo y -> SNo z -> SNo w -> SNo u -> SNo v -> SNo x2 -> SNo y2 -> (x + v) < x2 + y2 -> (y + y2) < u -> (x2 + z) < w + v -> (x + y + z) < w + u axiom add_SNo_com_3_0_1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = y + x + z const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoL : set set const SNoR : set set const SNoS_ : set set const SNoLev : 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 y hyp !y2:set.y2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * y2) -> (!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoL y2 -> (z2 * y2 + x * w2) < x * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoR y2 -> (z2 * y2 + x * w2) < x * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoR y2 -> (x * y2 + z2 * w2) < z2 * y2 + x * w2) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoL y2 -> (x * y2 + z2 * w2) < z2 * y2 + x * w2) -> P) -> P hyp !y2:set.y2 iIn SNoR x -> !z2:set.SNo z2 -> SNo (y2 * z2) hyp !y2:set.y2 iIn SNoL y -> SNo (x * y2) hyp u iIn SNoR x hyp v iIn SNoL 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 (u * w) hyp (z * y + u * w) < u * y + z * w hyp x2 iIn SNoR v hyp SNo x2 hyp x2 iIn SNoL y hyp (x * w + u * x2) < x * x2 + u * w claim (x * x2 + u * v) < x * v + u * x2 -> (z * y + x * w + u * v) < u * y + x * v + z * w