const SNo : set prop const In : set set prop term iIn = In infix iIn 2000 2000 const SNoLev : set set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const SNoR : set set axiom SNoR_I: !x:set.SNo x -> !y:set.SNo y -> SNoLev y iIn SNoLev x -> x < y -> y iIn SNoR x const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const SNoL : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> SNo z -> SNo w -> SNo (x * y) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL y -> (x2 * y + x * y2) < x * y + x2 * y2) -> SNo (z * y) -> (!x2:set.x2 iIn SNoR z -> !y2:set.y2 iIn SNoL y -> (z * y + x2 * y2) < x2 * y + z * y2) -> SNo (x * w) -> (!x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR w -> (x * w + x2 * y2) < x2 * w + x * y2) -> SNo (z * w) -> (!x2:set.x2 iIn SNoR z -> !y2:set.y2 iIn SNoR w -> (x2 * w + z * y2) < z * w + x2 * y2) -> SNo (z * y + x * w) -> SNo (x * y + z * w) -> SNo u -> u iIn SNoL x -> u iIn SNoR z -> SNo (u * y) -> SNo (u * w) -> SNo v -> w < v -> SNoLev v iIn SNoLev w -> v iIn SNoL y -> v iIn SNoR w -> (z * y + x * w) < x * y + z * w var x:set var y:set var z:set var w:set var u:set var v:set hyp SNo x hyp SNo y hyp SNo z hyp SNo w hyp SNo (x * y) hyp !x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoL y -> (x2 * y + x * y2) < x * y + x2 * y2 hyp SNo (z * y) hyp !x2:set.x2 iIn SNoR z -> !y2:set.y2 iIn SNoL y -> (z * y + x2 * y2) < x2 * y + z * y2 hyp SNo (x * w) hyp !x2:set.x2 iIn SNoL x -> !y2:set.y2 iIn SNoR w -> (x * w + x2 * y2) < x2 * w + x * y2 hyp SNo (z * w) hyp !x2:set.x2 iIn SNoR z -> !y2:set.y2 iIn SNoR w -> (x2 * w + z * y2) < z * w + x2 * y2 hyp SNo (z * y + x * w) hyp SNo (x * y + z * w) hyp SNo u hyp u iIn SNoL x hyp u iIn SNoR z hyp SNo (u * y) hyp SNo (u * w) hyp SNo v hyp w < v hyp v < y hyp SNoLev v iIn SNoLev w hyp SNoLev v iIn SNoLev y claim v iIn SNoL y -> (z * y + x * w) < x * y + z * w