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 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 minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x const SNoL : set set const SNoS_ : set set const SNoLev : set set axiom SNoL_SNoS: !x:set.SNo x -> !y:set.y iIn SNoL x -> y iIn SNoS_ (SNoLev x) const SNoR : set set axiom SNoR_SNoS: !x:set.SNo x -> !y:set.y iIn SNoR x -> y iIn SNoS_ (SNoLev x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom minus_add_SNo_distr_3: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> - (x + y + z) = - x + - y + - z 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 SNoR - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn z) -> u iIn SNoL x -> v iIn SNoR y -> w = u * y + x * v + - u * v -> SNo u -> SNo v -> - u iIn SNoR - x -> - w = (- u) * y + (- x) * v + - (- u) * v -> - w iIn z const SNoLt : set set prop term < = SNoLt infix < 2020 2020 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 SNoR - x -> !y2:set.y2 iIn SNoR y -> x2 * y + (- x) * y2 + - x2 * y2 iIn z hyp u iIn SNoL x hyp v iIn SNoR y hyp w = u * y + x * v + - u * v hyp SNo u hyp SNoLev u iIn SNoLev x hyp u < x hyp SNo v hyp SNo - u claim - u iIn SNoR - x -> - w iIn z