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 z -> SNo w -> 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) -> 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) -> SNo (z * y + x * w) -> SNo (x * y + z * w) -> z iIn SNoL x -> SNo u -> w < u -> SNoLev u iIn SNoLev w -> u iIn SNoL y -> u 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 hyp SNo x hyp SNo y hyp SNo z hyp SNo w 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 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 SNo (z * y + x * w) hyp SNo (x * y + z * w) hyp z iIn SNoL x hyp SNo u hyp w < u hyp u < y hyp SNoLev u iIn SNoLev w hyp SNoLev u iIn SNoLev y claim u iIn SNoL y -> (z * y + x * w) < x * y + z * w