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 SNoLev : set set const minus_SNo : set set term - = minus_SNo 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_contra2: !x:set.!y:set.SNo x -> SNo y -> x < - y -> y < - x const SNoR : set set axiom SNoR_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> x < y -> y iIn SNoR x const SNoS_ : set set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoL : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 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 SNoR x -> !x2:set.x2 iIn SNoL y -> v * y + x * x2 + - v * x2 iIn z) -> u iIn SNoL y -> SNo w -> SNoLev w iIn SNoLev - x -> w < - x -> SNo u -> SNo - w -> - w iIn SNoR x -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo 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 SNoR 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 w < - x hyp SNo u claim SNo - w -> w * y + (- x) * u + - w * u iIn Repl z minus_SNo