Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: atleastp x0 x1.

Assume H1: atleastp x1 x2.

Apply H0 with atleastp x0 x2.

Let x3 of type ι → ι be given.

Assume H2: inj x0 x1 x3.

Apply H1 with atleastp x0 x2.

Let x4 of type ι → ι be given.

Assume H3: inj x1 x2 x4.

Let x5 of type ο be given.

Assume H4: ∀ x6 : ι → ι . inj x0 x2 x6 ⟶ x5.

Apply H4 with λ x6 . x4 (x3 x6).

Apply inj_comp with x0, x1, x2, x3, x4 leaving 2 subgoals.

The subproof is completed by applying H2.

The subproof is completed by applying H3.

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Proofgold Proof