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 mul_SNo : set set set term * = mul_SNo infix * 2291 2290 axiom SNo_mul_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x * y) const SNoR : set set const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P const SNoL : set set 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 add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> SNo z -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR y -> (z + x2 * y2) < x2 * y + x * y2) -> SNo w -> w < z -> u iIn SNoL x -> v iIn SNoR y -> (u * y + x * v) <= w + u * v -> SNo u -> SNo v -> SNo (u * v) -> w < x * y lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo y -> SNo z -> (!x2:set.x2 iIn SNoR x -> !y2:set.y2 iIn SNoL y -> (z + x2 * y2) < x2 * y + x * y2) -> SNo w -> w < z -> u iIn SNoR x -> v iIn SNoL y -> (u * y + x * v) <= w + u * v -> SNo u -> SNo v -> SNo (u * v) -> w < x * y const SNoS_ : set set var x:set var y:set var z:set var w:set hyp SNo x hyp SNo y hyp SNo (x * y) hyp SNo z hyp !u:set.u iIn SNoS_ (SNoLev z) -> SNoLev u iIn SNoLev (x * y) -> x * y < u -> (?v:set.v iIn SNoL x & ?x2:set.x2 iIn SNoR y & (v * y + x * x2) <= u + v * x2) | ?v:set.v iIn SNoR x & ?x2:set.x2 iIn SNoL y & (v * y + x * x2) <= u + v * x2 hyp SNoLev z iIn SNoLev (x * y) hyp !u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> (z + u * v) < u * y + x * v hyp !u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> (z + u * v) < u * y + x * v hyp SNo w hyp SNoLev w iIn SNoLev z hyp w < z hyp x * y < w claim (?u:set.u iIn SNoL x & ?v:set.v iIn SNoR y & (u * y + x * v) <= w + u * v) | (?u:set.u iIn SNoR x & ?v:set.v iIn SNoL y & (u * y + x * v) <= w + u * v) -> w < x * y