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.SNo x -> SNo y -> SNo z -> SNo w -> w < y -> SNo (x * y) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> (v * y + x * x2) < x * y + v * x2) -> SNo (z * y) -> (!v:set.v iIn SNoR z -> !x2:set.x2 iIn SNoL y -> (z * y + v * x2) < v * y + z * x2) -> SNo (x * w) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR w -> (x * w + v * x2) < v * w + x * x2) -> SNo (z * w) -> (!v:set.v iIn SNoR z -> !x2:set.x2 iIn SNoR w -> (v * w + z * x2) < z * w + v * x2) -> SNo (z * y + x * w) -> SNo (x * y + z * w) -> SNo u -> z < u -> SNoLev u iIn SNoLev z -> u iIn SNoL x -> u iIn SNoR z -> (z * y + x * w) < x * y + z * w var x:set var y:set var z:set var w:set var u:set hyp SNo x hyp SNo y hyp SNo z hyp SNo w hyp w < y hyp SNo (x * y) hyp !v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> (v * y + x * x2) < x * y + v * x2 hyp SNo (z * y) hyp !v:set.v iIn SNoR z -> !x2:set.x2 iIn SNoL y -> (z * y + v * x2) < v * y + z * x2 hyp SNo (x * w) hyp !v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR w -> (x * w + v * x2) < v * w + x * x2 hyp SNo (z * w) hyp !v:set.v iIn SNoR z -> !x2:set.x2 iIn SNoR w -> (v * w + z * x2) < z * w + v * x2 hyp SNo (z * y + x * w) hyp SNo (x * y + z * w) hyp SNo u hyp z < u hyp u < x hyp SNoLev u iIn SNoLev z hyp SNoLev u iIn SNoLev x claim u iIn SNoL x -> (z * y + x * w) < x * y + z * w