const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoLev : set set axiom minus_SNo_Lev: !x:set.SNo x -> SNoLev - x = SNoLev x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom minus_SNo_Lt_contra: !x:set.!y:set.SNo x -> SNo y -> x < y -> - y < - x 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 const SNoS_ : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoR : set set 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.SNo x -> SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> (- x2) * y = - x2 * y) -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> (- x) * x2 = - x * x2) -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> (- x2) * y2 = - x2 * y2) -> (!x2:set.x2 iIn SNoL - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn z) -> u iIn SNoR x -> v iIn SNoR y -> w = u * y + x * v + - u * v -> SNo u -> SNoLev u iIn SNoLev x -> x < u -> SNo v -> SNo - u -> - u iIn SNoL - x -> - w iIn z var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp SNo y hyp !x2:set.x2 iIn SNoS_ (SNoLev x) -> (- x2) * y = - x2 * y hyp !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- x) * x2 = - x * x2 hyp !x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev y) -> (- x2) * y2 = - x2 * y2 hyp !x2:set.x2 iIn SNoL - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn z hyp u iIn SNoR x hyp v iIn SNoR y hyp w = u * y + x * v + - u * v hyp SNo u hyp SNoLev u iIn SNoLev x hyp x < u hyp SNo v claim SNo - u -> - w iIn z