const famunion : set (set set) set const Repl : set (set set) set const setsum : set set set term Sigma = \x:set.\f:set set.famunion x \y:set.Repl (f y) (setsum y) const In : set set prop term iIn = In infix iIn 2000 2000 axiom famunionE: !x:set.!f:set set.!y:set.y iIn famunion x f -> ?z:set.z iIn x & y iIn f z const proj0 : set set const proj1 : set set lemma !x:set.!f:set set.!y:set.y iIn Sigma x f -> (?z:set.z iIn x & y iIn Repl (f z) (setsum z)) -> setsum (proj0 y) (proj1 y) = y & proj0 y iIn x & proj1 y iIn f (proj0 y) claim !x:set.!f:set set.!y:set.y iIn Sigma x f -> setsum (proj0 y) (proj1 y) = y & proj0 y iIn x & proj1 y iIn f (proj0 y)