Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Let x3 of type ι be given.

Let x4 of type ι be given.

Assume H0: Field_Hom x0 x1 x3.

Assume H1: Field_Hom x1 x2 x4.

Apply Field_Hom_E with x0, x1, x3, Field_Hom x0 x2 (lam_comp (ap x0 0) x4 x3) leaving 2 subgoals.

The subproof is completed by applying H0.

Assume H2: Field x0.

Assume H3: Field x1.

Assume H4: x3 ∈ setexp (field0 x1) (field0 x0).

Assume H5: ap x3 (field3 x0) = field3 x1.

Assume H6: ap x3 (field4 x0) = field4 x1.

Assume H7: ∀ x5 . x5 ∈ field0 x0 ⟶ ∀ x6 . x6 ∈ field0 x0 ⟶ ap x3 (field1b x0 x5 x6) = field1b x1 (ap x3 x5) (ap x3 x6).

Assume H8: ∀ x5 . x5 ∈ field0 x0 ⟶ ∀ x6 . x6 ∈ field0 x0 ⟶ ap x3 (field2b x0 x5 x6) = field2b x1 (ap x3 x5) (ap x3 x6).

Assume H9: ∀ x5 . x5 ∈ field0 x0 ⟶ ap x3 (Field_minus x0 x5) = Field_minus x1 (ap x3 x5).

Assume H10: ∀ x5 . x5 ∈ field0 x0 ⟶ ap x3 x5 = field3 x1 ⟶ x5 = field3 x0.

Assume H11: ∀ x5 . x5 ∈ field0 x0 ⟶ ∀ x6 . x6 ∈ field0 x0 ⟶ ap x3 x5 = ap x3 x6 ⟶ x5 = x6.

Assume H12: ∀ x5 . x5 ∈ field0 x0 ⟶ ∀ x6 . x6 ∈ omega ⟶ ap x3 (CRing_with_id_omega_exp x0 x5 x6) = CRing_with_id_omega_exp x1 (ap x3 x5) x6.

Apply Field_Hom_E with x1, x2, x4, Field_Hom x0 x2 (lam_comp (ap x0 0) x4 x3) leaving 2 subgoals.

The subproof is completed by applying H1.

Assume H13: Field x1.

Assume H14: Field x2.

Assume H15: x4 ∈ setexp (field0 x2) (field0 x1).

Assume H16: ap x4 (field3 x1) = field3 x2.

Assume H17: ap x4 (field4 x1) = field4 x2.

Assume H18: ∀ x5 . x5 ∈ field0 x1 ⟶ ∀ x6 . x6 ∈ field0 x1 ⟶ ap x4 (field1b x1 x5 x6) = field1b x2 (ap x4 x5) (ap x4 x6).

Assume H19: ∀ x5 . x5 ∈ field0 x1 ⟶ ∀ x6 . x6 ∈ field0 x1 ⟶ ap x4 (field2b x1 x5 x6) = field2b x2 (ap x4 x5) (ap x4 x6).

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Proofgold Proof