const bij : set set (set set) prop term equip = \x:set.\y:set.?f:set set.bij x y f const In : set set prop term iIn = In infix iIn 2000 2000 term PNoEq_ = \x:set.\p:set prop.\q:set prop.!y:set.y iIn x -> (p y <-> q y) term nIn = \x:set.\y:set.~ x iIn y term SNoEq_ = \x:set.\y:set.\z:set.PNoEq_ x (\w:set.w iIn y) \w:set.w iIn z const SNo : set prop const SNoLev : set set const SNo_extend0 : set set axiom SNo_extend0_nIn: !x:set.SNo x -> nIn (SNoLev x) (SNo_extend0 x) const nat_p : set prop const exp_SNo_nat : set set set const ordsucc : set set const Empty : set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const binintersect : set set set const SNoElts_ : set set const minus_SNo : set set term - = minus_SNo const Sep : set (set prop) set const SNoS_ : set set const omega : set lemma !x:set.!f:set set.!f2:set set.!y:set.!z:set.nat_p x -> SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x) -> (!w:set.nIn x w -> f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x))) -> SNo y -> f z = y + - exp_SNo_nat (ordsucc (ordsucc Empty)) x -> SNoLev z = x -> SNo z -> SNoLev (SNo_extend0 z) = ordsucc x -> nIn x (SNo_extend0 z) -> ?w:set.w iIn Sep (SNoS_ omega) (\u:set.SNoLev u = ordsucc x) & f2 w = y var x:set var f:set set var f2:set set var y:set var z:set hyp nat_p x hyp SNo (exp_SNo_nat (ordsucc (ordsucc Empty)) x) hyp !w:set.nIn x w -> f2 w = exp_SNo_nat (ordsucc (ordsucc Empty)) x + f (binintersect w (SNoElts_ x)) hyp SNo y hyp f z = y + - exp_SNo_nat (ordsucc (ordsucc Empty)) x hyp SNoLev z = x hyp SNo z claim SNoLev (SNo_extend0 z) = ordsucc x -> ?w:set.w iIn Sep (SNoS_ omega) (\u:set.SNoLev u = ordsucc x) & f2 w = y