Apply nat_ind with λ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ setminus omega 1) ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⊆ nat_primrec 1 (λ x3 x4 . mul_nat (x1 x3) x4) x0 leaving 2 subgoals.

Let x0 of type ι → ι be given.

Assume H0: ∀ x1 . x1 ∈ 0 ⟶ x0 x1 ∈ setminus omega 1.

Let x1 of type ι be given.

Assume H1: x1 ∈ 0.

Apply FalseE with x0 x1 ⊆ nat_primrec 1 (λ x2 x3 . mul_nat (x0 x2) x3) 0.

Apply EmptyE with x1.

The subproof is completed by applying H1.

Let x0 of type ι be given.

Assume H0: nat_p x0.

Assume H1: ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ setminus omega 1) ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⊆ nat_primrec 1 (λ x3 x4 . mul_nat (x1 x3) x4) x0.

Let x1 of type ι → ι be given.

Assume H2: ∀ x2 . x2 ∈ ordsucc x0 ⟶ x1 x2 ∈ setminus omega 1.

Let x2 of type ι be given.

Assume H3: x2 ∈ ordsucc x0.

Claim L4: ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ∈ setminus omega 1

Let x3 of type ι be given.

Assume H4: x3 ∈ x0.

Apply H2 with x3.

Apply ordsuccI1 with x0, x3.

The subproof is completed by applying H4.

Claim L5: ∀ x3 . x3 ∈ ordsucc x0 ⟶ nat_p (x1 x3)

Let x3 of type ι be given.

Assume H5: x3 ∈ ordsucc x0.

Apply omega_nat_p with x1 x3.

Apply setminusE1 with omega, 1, x1 x3.

Apply H2 with x3.

The subproof is completed by applying H5.

Claim L6: ∀ x3 . x3 ∈ x0 ⟶ nat_p (x1 x3)

Let x3 of type ι be given.

Assume H6: x3 ∈ x0.

Apply L5 with x3.

Apply ordsuccI1 with x0, x3.

The subproof is completed by applying H6.

Apply nat_primrec_S with 1, λ x3 x4 . mul_nat (x1 x3) x4, x0, λ x3 x4 . x1 x2 ⊆ x4 leaving 2 subgoals.

The subproof is completed by applying H0.

Apply ordsuccE with x0, x2, x1 x2 ⊆ mul_nat (x1 x0) (nat_primrec 1 (λ x3 x4 . mul_nat (x1 x3) x4) x0) leaving 3 subgoals.

The subproof is completed by applying H3.

Assume H7: x2 ∈ x0.

Apply Subq_tra with x1 x2, nat_primrec 1 (λ x3 x4 . mul_nat (x1 x3) x4) x0, mul_nat (x1 x0) (nat_primrec 1 (λ x3 x4 . mul_nat (x1 x3) x4) x0) leaving 2 subgoals.

Apply H1 with x1, x2 leaving 2 subgoals.

The subproof is completed by applying L4.

The subproof is completed by applying H7.

Apply unknownprop_97c7776a66590be2b2527dd1cda3d7d16f5d5fe279216d0058fa4bf744a721e1 with nat_primrec 1 (λ x3 x4 . mul_nat (x1 x3) x4) x0, x1 x0 leaving 2 subgoals.

Apply unknownprop_6acf4d775f3657f6657248067c81d93f75e7e3c111f6937130fc5b44c841c89f with x0, x1 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying L6.

Apply H2 with x0.

The subproof is completed by applying ordsuccI2 with x0.

Assume H7: x2 = x0.

Apply H7 with λ x3 x4 . x1 x4 ⊆ mul_nat (x1 x0) (nat_primrec 1 (λ x5 x6 . mul_nat (x1 x5) x6) x0).

Apply unknownprop_0dc8d11d1ba28645d1565e6f95fe26f514da291413e114d0327c09556f7d23e9 with x1 x0, nat_primrec 1 (λ x3 x4 . mul_nat (x1 x3) x4) x0 leaving 2 subgoals.

Apply L5 with x0.

The subproof is completed by applying ordsuccI2 with x0.

Apply unknownprop_50b65292d8c21d0a51ffd7239891113269f96ca6881e5f223c17754525c2ffd2 with x0, x1 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying L4.

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Proofgold Proof