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 ι be given.
Let x4 of type ι → ι → ι be given.
Apply explicit_Nats_E with
x0,
x1,
x2,
explicit_Nats_primrec x0 x1 x2 x3 x4 x1 = x3.
Assume H3:
∀ x5 . prim1 x5 x0 ⟶ x2 x5 = x1 ⟶ ∀ x6 : ο . x6.
Assume H4:
∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 = x2 x6 ⟶ x5 = x6.
Assume H5:
∀ x5 : ι → ο . x5 x1 ⟶ (∀ x6 . x5 x6 ⟶ x5 (x2 x6)) ⟶ ∀ x6 . prim1 x6 x0 ⟶ x5 x6.
Apply unknownprop_987d3840aa104d50ea50759bc446be3aae0e33c59dc8291c7942424d9287e6ed with
x0,
x1,
x2,
x3,
x4,
explicit_Nats_primrec x0 x1 x2 x3 x4 x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply unknownprop_babb8ca41ea0201511f2f22263861b8dd90f74016344bcbe2499d80c10dc00c2 with
x0,
x1,
x2,
x3,
x4,
x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.