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 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 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) axiom minus_SNo_invol: !x:set.SNo x -> - - x = x const Repl : set (set set) set lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo x -> SNo y -> (!v:set.v iIn SNoS_ (SNoLev x) -> (- v) * y = - v * y) -> (!v:set.v iIn SNoS_ (SNoLev y) -> (- x) * v = - x * v) -> (!v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- v) * x2 = - v * x2) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> v * y + x * x2 + - v * x2 iIn z) -> u iIn SNoL y -> SNo w -> SNo u -> SNo - w -> - w iIn SNoL x -> w * y + (- x) * u + - w * u = - ((- w) * y + x * u + - (- w) * u) -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo 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 hyp SNo x hyp SNo y hyp !v:set.v iIn SNoS_ (SNoLev x) -> (- v) * y = - v * y hyp !v:set.v iIn SNoS_ (SNoLev y) -> (- x) * v = - x * v hyp !v:set.v iIn SNoS_ (SNoLev x) -> !x2:set.x2 iIn SNoS_ (SNoLev y) -> (- v) * x2 = - v * x2 hyp !v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> v * y + x * x2 + - v * x2 iIn z hyp u iIn SNoL y hyp SNo w hyp SNoLev w iIn SNoLev - x hyp - x < w hyp SNo u hyp SNo - w claim - w iIn SNoL x -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo