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 const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const SNoR : set set const SNoLev : 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 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const minus_SNo : set set term - = minus_SNo lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.SNo x -> SNo y -> SNo (x * y) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoR y -> (z2 * y + x * w2) < x * y + z2 * w2) -> (!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoR y -> (x * y + z2 * w2) < z2 * y + x * w2) -> u iIn SNoR x -> v iIn SNoR y -> z = u * y + x * v + - u * v -> SNo (u * y) -> SNo (x * v) -> SNo (u * v) -> x2 iIn SNoL x -> y2 iIn SNoR y -> w = x2 * y + x * y2 + - x2 * y2 -> SNo x2 -> SNo y2 -> SNo (x2 * y) -> z < w lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.SNo x -> SNo y -> SNo (x * y) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoR y -> (z2 * y + x * w2) < x * y + z2 * w2) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoL y -> (x * y + z2 * w2) < z2 * y + x * w2) -> u iIn SNoR x -> v iIn SNoR y -> z = u * y + x * v + - u * v -> SNo (u * y) -> SNo (x * v) -> SNo (u * v) -> x2 iIn SNoR x -> y2 iIn SNoL y -> w = x2 * y + x * y2 + - x2 * y2 -> SNo x2 -> SNo y2 -> SNo (x2 * y) -> z < w 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 SNo y hyp !y2:set.y2 iIn z -> !P:prop.(!z2:set.z2 iIn SNoL x -> !w2:set.w2 iIn SNoR y -> y2 = z2 * y + x * w2 + - z2 * w2 -> P) -> (!z2:set.z2 iIn SNoR x -> !w2:set.w2 iIn SNoL y -> y2 = z2 * y + x * w2 + - z2 * w2 -> P) -> P hyp SNo (x * y) hyp !y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR y -> (y2 * y + x * z2) < x * y + y2 * z2 hyp !y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR y -> (x * y + y2 * z2) < y2 * y + x * z2 hyp !y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL y -> (x * y + y2 * z2) < y2 * y + x * z2 hyp u iIn z hyp v iIn SNoR x hyp x2 iIn SNoR y hyp w = v * y + x * x2 + - v * x2 hyp SNo v hyp SNo x2 hyp SNo (v * y) hyp SNo (x * x2) claim SNo (v * x2) -> w < u