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 nIn = \x:set.\y:set.~ x iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const SNo : set prop const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const minus_SNo : set set term - = minus_SNo const omega : set const ap : set set set const SNoCut : set set set const Repl : set (set set) set const SNoLt : set set prop term < = SNoLt infix < 2020 2020 const add_SNo : set set set term + = add_SNo infix + 2281 2280 const eps_ : set set const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.!z2:set.SNo x -> SNo y -> SNo (x * y) -> SNo - x * y -> (!w2:set.w2 iIn omega -> SNo (ap w w2)) -> SNo (SNoCut (Repl omega (ap z)) (Repl omega (ap w))) -> (!w2:set.w2 iIn omega -> SNoCut (Repl omega (ap z)) (Repl omega (ap w)) < ap w w2) -> (!w2:set.w2 iIn omega -> (ap w w2 + - eps_ w2) < x * y) -> u = v * y + x * x2 + - v * x2 -> y2 iIn omega -> eps_ y2 <= v + - x -> z2 iIn omega -> eps_ z2 <= y + - x2 -> SNo v -> SNo x2 -> SNo (eps_ y2) -> SNo (eps_ z2) -> y2 + z2 iIn omega -> SNo (eps_ (y2 + z2)) -> SNo - eps_ (y2 + z2) -> SNo (ap w (y2 + z2)) -> SNoCut (Repl omega (ap z)) (Repl omega (ap w)) < u var x:set var y:set var z:set var w:set var u:set var v:set var x2:set var y2:set var z2:set hyp SNo x hyp SNo y hyp SNo (x * y) hyp SNo - x * y hyp !w2:set.w2 iIn omega -> SNo (ap w w2) hyp SNo (SNoCut (Repl omega (ap z)) (Repl omega (ap w))) hyp !w2:set.w2 iIn omega -> SNoCut (Repl omega (ap z)) (Repl omega (ap w)) < ap w w2 hyp !w2:set.w2 iIn omega -> (ap w w2 + - eps_ w2) < x * y hyp u = v * y + x * x2 + - v * x2 hyp y2 iIn omega hyp eps_ y2 <= v + - x hyp z2 iIn omega hyp eps_ z2 <= y + - x2 hyp SNo v hyp SNo x2 hyp SNo (eps_ y2) hyp SNo (eps_ z2) hyp y2 + z2 iIn omega hyp SNo (eps_ (y2 + z2)) claim SNo - eps_ (y2 + z2) -> SNoCut (Repl omega (ap z)) (Repl omega (ap w)) < u