Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: Field_Hom x0 x1 x2.

Apply Field_Hom_E with x0, x1, x2, ∀ x3 . x3 ∈ field0 x0 ⟶ ∀ x4 . x4 ∈ field0 x0 ⟶ ap x2 x3 = ap x2 x4 ⟶ x3 = x4 leaving 2 subgoals.

The subproof is completed by applying H0.

Assume H1: Field x0.

Assume H2: Field x1.

Assume H3: x2 ∈ setexp (field0 x1) (field0 x0).

Assume H4: ap x2 (field3 x0) = field3 x1.

Assume H5: ap x2 (field4 x0) = field4 x1.

Assume H6: ∀ x3 . x3 ∈ field0 x0 ⟶ ∀ x4 . x4 ∈ field0 x0 ⟶ ap x2 (field1b x0 x3 x4) = field1b x1 (ap x2 x3) (ap x2 x4).

Assume H7: ∀ x3 . x3 ∈ field0 x0 ⟶ ∀ x4 . x4 ∈ field0 x0 ⟶ ap x2 (field2b x0 x3 x4) = field2b x1 (ap x2 x3) (ap x2 x4).

Assume H8: ∀ x3 . x3 ∈ field0 x0 ⟶ ap x2 (Field_minus x0 x3) = Field_minus x1 (ap x2 x3).

Assume H9: ∀ x3 . x3 ∈ field0 x0 ⟶ ap x2 x3 = field3 x1 ⟶ x3 = field3 x0.

Assume H10: ∀ x3 . x3 ∈ field0 x0 ⟶ ∀ x4 . x4 ∈ field0 x0 ⟶ ap x2 x3 = ap x2 x4 ⟶ x3 = x4.

Assume H11: ∀ x3 . x3 ∈ field0 x0 ⟶ ∀ x4 . x4 ∈ omega ⟶ ap x2 (CRing_with_id_omega_exp x0 x3 x4) = CRing_with_id_omega_exp x1 (ap x2 x3) x4.

The subproof is completed by applying H10.

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Proofgold Proof