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) const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom add_SNo_Lt2: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> y < z -> (x + y) < x + z axiom add_SNo_rotate_3_1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = z + x + y axiom add_SNo_Lt1: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < z -> (x + y) < z + y axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z axiom SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z axiom add_SNo_Lt2_cancel: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) < x + z -> y < z const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 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 u iIn SNoR z hyp SNo (u * y) hyp SNo (u * w) hyp SNo (z * w + u * y) hyp SNo (u * w + x * y) hyp SNo (x * w + u * y) hyp SNo (u * w + z * y) hyp y iIn SNoR w hyp (x * w + u * y) < u * w + x * y claim (u * w + z * y) < z * w + u * y -> (z * y + x * w) < x * y + z * w