Let x0 of type ι be given.

Let x1 of type ι → ι → ι be given.

Let x2 of type ι → ι → ι be given.

Assume H0: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4.

Claim L2: ...

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Apply H1 with ∀ x3 . x3 ∈ x0 ⟶ explicit_Group_inverse x0 x1 x3 = explicit_Group_inverse x0 x2 x3.

Assume H3: and (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 ∈ x0) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x3 (x1 x4 x5) = x1 (x1 x3 x4) x5).

Assume H4: ∃ x3 . and (x3 ∈ x0) (and (∀ x4 . x4 ∈ x0 ⟶ and (x1 x3 x4 = x4) (x1 x4 x3 = x4)) (∀ x4 . x4 ∈ x0 ⟶ ∃ x5 . and (x5 ∈ x0) (and (x1 x4 x5 = x3) (x1 x5 x4 = x3)))).

Apply H3 with ∀ x3 . x3 ∈ x0 ⟶ explicit_Group_inverse x0 x1 x3 = explicit_Group_inverse x0 x2 x3.

Assume H5: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 ∈ x0.

Assume H6: ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x3 (x1 x4 x5) = x1 (x1 x3 x4) x5.

Claim L7: ...

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Claim L8: ...

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Claim L9: ...

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Let x3 of type ι be given.

Assume H10: x3 ∈ x0.

Claim L11: ...

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Claim L12: ...

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Claim L13: ...

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Claim L14: ...

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Apply L9 with explicit_Group_inverse x0 x2 x3, λ x4 x5 . explicit_Group_inverse x0 x1 x3 = x4 leaving 2 subgoals.

The subproof is completed by applying L13.

Apply H0 with explicit_Group_identity x0 x2, explicit_Group_inverse x0 x2 x3, λ x4 x5 . explicit_Group_inverse x0 x1 x3 = x4 leaving 3 subgoals.

The subproof is completed by applying L8.

The subproof is completed by applying L13.

Apply explicit_Group_identity_repindep with x0, x1, x2, λ x4 x5 . explicit_Group_inverse x0 x1 x3 = x1 x4 (explicit_Group_inverse x0 x2 x3) leaving 3 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Apply L12 with λ x4 x5 . explicit_Group_inverse x0 x1 x3 = x1 x4 (explicit_Group_inverse x0 x2 x3).

Apply H6 with explicit_Group_inverse x0 x1 x3, x3, explicit_Group_inverse x0 x2 x3, λ x4 x5 . explicit_Group_inverse x0 x1 x3 = x4 leaving 4 subgoals.

The subproof is completed by applying L11.

The subproof is completed by applying H10.

The subproof is completed by applying L13.

Apply H0 with x3, explicit_Group_inverse x0 x2 x3, λ x4 x5 . explicit_Group_inverse x0 x1 x3 = x1 (explicit_Group_inverse x0 x1 x3) x5 leaving 3 subgoals.

The subproof is completed by applying H10.

The subproof is completed by applying L13.

Apply L14 with λ x4 x5 . explicit_Group_inverse x0 x1 x3 = x1 (explicit_Group_inverse x0 x1 x3) x5.

Apply explicit_Group_identity_repindep with x0, x1, x2, λ x4 x5 . explicit_Group_inverse x0 x1 ... = ... leaving 3 subgoals.

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Proofgold Proof