const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom add_SNo_Lt1_cancel: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) < z + y -> x < z const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 lemma !x:set.!y:set.!z:set.!w:set.!u:set.SNo (x * y) -> SNo (z * y) -> SNo (x * w) -> SNo (z * w) -> SNo (x * u) -> SNo (z * u) -> (z * y + x * u) < x * y + z * u -> (x * w + z * u) < z * w + x * u -> SNo (z * y + x * u) -> (x * w + z * y + x * u) < x * y + z * w + x * u const In : set set prop term iIn = In infix iIn 2000 2000 const SNoR : set set var x:set var y:set var z:set var w:set var u:set hyp SNo (x * y) hyp SNo (z * y) hyp SNo (x * w) hyp SNo (z * w) hyp !v:set.v iIn SNoR z -> !x2:set.x2 iIn SNoR w -> (v * w + z * x2) < z * w + v * x2 hyp SNo (z * y + x * w) hyp SNo (x * y + z * w) hyp x iIn SNoR z hyp u iIn SNoR w hyp SNo (x * u) hyp SNo (z * u) hyp (z * y + x * u) < x * y + z * u claim (x * w + z * u) < z * w + x * u -> (z * y + x * w) < x * y + z * w