const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const SNo : set prop const SNoL : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoR : set set var x:set var y:set hyp SNo x hyp SNo y hyp SNo (x * y) claim (!z:set.SNo z -> SNoLev z iIn SNoLev (x * y) -> z < x * y -> (?w:set.w iIn SNoL x & ?u:set.u iIn SNoL y & (z + w * u) <= w * y + x * u) | ?w:set.w iIn SNoR x & ?u:set.u iIn SNoR y & (z + w * u) <= w * y + x * u) -> !z:set.z iIn SNoL (x * y) -> (?w:set.w iIn SNoL x & ?u:set.u iIn SNoL y & (z + w * u) <= w * y + x * u) | ?w:set.w iIn SNoR x & ?u:set.u iIn SNoR y & (z + w * u) <= w * y + x * u