const In : set set prop term iIn = In infix iIn 2000 2000 term nIn = \x:set.\y:set.~ x iIn y const Empty : set axiom EmptyE: !x:set.nIn x Empty axiom FalseE: ~ False const ordsucc : set set axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x const nat_p : set prop axiom nat_ind: !p:set prop.p Empty -> (!x:set.nat_p x -> p x -> p (ordsucc x)) -> !x:set.nat_p x -> p x lemma !p:set prop.(!x:set.nat_p x -> (!y:set.y iIn x -> p y) -> p x) -> (!x:set.nat_p x -> !y:set.y iIn x -> p y) -> !x:set.nat_p x -> p x claim !p:set prop.(!x:set.nat_p x -> (!y:set.y iIn x -> p y) -> p x) -> !x:set.nat_p x -> p x