const If_i : prop set set set axiom If_i_0: !P:prop.!x:set.!y:set.~ P -> If_i P x y = y axiom xm: !P:prop.P | ~ P const In : set set prop term iIn = In infix iIn 2000 2000 const Union : set set lemma !x:set.!g:set set set.!y:set.!f:set set.!f2:set set.~ Union y iIn y -> If_i (Union y iIn y) (g (Union y) (f (Union y))) x = x -> If_i (Union y iIn y) (g (Union y) (f (Union y))) x = If_i (Union y iIn y) (g (Union y) (f2 (Union y))) x claim !x:set.!g:set set set.!y:set.!f:set set.!f2:set set.(!z:set.z iIn y -> f z = f2 z) -> If_i (Union y iIn y) (g (Union y) (f (Union y))) x = If_i (Union y iIn y) (g (Union y) (f2 (Union y))) x