Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι → ι be given.
Let x3 of type ο be given.
Assume H0:
explicit_Nats x0 x1 x2 ⟶ prim1 x1 x0 ⟶ (∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0) ⟶ (∀ x4 . prim1 x4 x0 ⟶ x2 x4 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 : ι → ο . x4 x1 ⟶ (∀ x5 . x4 x5 ⟶ x4 (x2 x5)) ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x5) ⟶ x3.
Apply and5E with
prim1 x1 x0,
∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0,
∀ x4 . prim1 x4 x0 ⟶ x2 x4 = x1 ⟶ ∀ x5 : ο . x5,
∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5,
∀ x4 : ι → ο . x4 x1 ⟶ (∀ x5 . x4 x5 ⟶ x4 (x2 x5)) ⟶ ∀ x5 . prim1 x5 x0 ⟶ x4 x5,
x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Apply H0.
The subproof is completed by applying H1.