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_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x 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 SNoLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x < y -> y < z -> x < z const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 var x:set var y:set var z:set var w:set var u:set hyp SNo (x * y) hyp SNo (x * w) hyp SNo (z * w) hyp SNo (x * u) hyp SNo (z * u) hyp (z * y + x * u) < x * y + z * u hyp (x * w + z * u) < z * w + x * u hyp SNo (z * y + x * u) hyp SNo (z * u + x * y) claim SNo (z * w + x * u) -> (x * w + z * y + x * u) < x * y + z * w + x * u