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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ιι be given.
Apply explicit_Nats_E with x0, x1, x2, ∀ x3 . x3x0∀ x4 . x4x0explicit_Nats_one_exp x0 x1 x2 x3 x4x0.
Assume H0: explicit_Nats x0 x1 x2.
Assume H1: x1x0.
Assume H2: ∀ x3 . x3x0x2 x3x0.
Assume H3: ∀ x3 . x3x0x2 x3 = x1∀ x4 : ο . x4.
Assume H4: ∀ x3 . x3x0∀ x4 . x4x0x2 x3 = x2 x4x3 = x4.
Assume H5: ∀ x3 : ι → ο . x3 x1(∀ x4 . x3 x4x3 (x2 x4))∀ x4 . x4x0x3 x4.
Let x3 of type ι be given.
Assume H6: x3x0.
Let x4 of type ι be given.
Assume H7: x4x0.
Apply explicit_Nats_primrec_P with x0, x1, x2, λ x5 . x5x0, x3, λ x5 x6 . explicit_Nats_one_mult x0 x1 x2 x3 x6, x4 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
Let x5 of type ι be given.
Assume H8: x5x0.
Let x6 of type ι be given.
Assume H9: (λ x7 . x7x0) x6.
Apply explicit_Nats_one_mult_N with x0, x1, x2, x3, x6 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H6.
The subproof is completed by applying H9.
The subproof is completed by applying H7.