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 mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom mul_SNo_distrL: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x * (y + z) = x * y + x * z axiom mul_SNo_distrR: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> (x + y) * z = x * z + y * z axiom SNo_mul_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_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 claim !x:set.!y:set.!z:set.!w:set.SNo x -> SNo y -> SNo z -> SNo w -> (x + y) * (z + w) = x * z + x * w + y * z + y * w