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_Lt3: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> x < z -> y < w -> (x + y) < z + w axiom add_SNo_com_4_inner_mid: !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (x + y) + z + w = (x + z) + y + w axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x 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 var v:set hyp SNo (x * y) hyp SNo (z * y) hyp SNo (x * w) hyp SNo (z * w) hyp SNo (z * y + x * w) hyp SNo (x * y + z * w) hyp SNo (u * y) hyp SNo (u * w) hyp SNo (x * v) hyp SNo (z * v) hyp SNo (u * v) hyp (u * y + x * v) < x * y + u * v hyp (u * w + z * v) < z * w + u * v hyp (x * w + u * v) < u * w + x * v hyp (z * y + u * v) < u * y + z * v hyp SNo (u * v + u * v) hyp SNo (u * y + z * v) hyp SNo (u * w + x * v) hyp SNo (z * y + u * v) hyp SNo (x * w + u * v) hyp SNo (u * w + z * v) hyp SNo (u * y + x * v) hyp SNo (x * y + u * v) claim SNo (z * w + u * v) -> ((z * y + x * w) + u * v + u * v) < (x * y + z * w) + u * v + u * v