Let X and Y be given.
Set R to be the term
(λX : set ⇒ λY : set ⇒ ∃f : set → set, (∀x ∈ X, In_rec_i_G x (f x)) ∧ Y = F X f).
Apply (H1 R) to the current goal.
We will
prove ∀Z : set, ∀g : set → set, (∀z ∈ Z, R z (g z)) → R Z (F Z g).
Let Z and g be given.
We use g to witness the existential quantifier.
Apply andI to the current goal.
Let z be given.
Let f of type set → set be given.
rewrite the current goal using H4 (from left to right).
An
exact proof term for the current goal is
(In_rec_i_G_c z f H3).
∎