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Proofgold Proof

pf
Let x0 of type (ιι) → ο be given.
Assume H0: ∃ x1 : ι → ι . x0 x1.
Assume H1: ∀ x1 x2 : ι → ι . x0 x1x0 x2x1 = x2.
Apply H0 with x0 (Descr_ii x0).
Let x1 of type ιι be given.
Assume H2: x0 x1.
Claim L3: x1 = Descr_ii x0
Apply functional extensionality with x1, Descr_ii x0.
Let x2 of type ι be given.
Claim L3: ∀ x3 : ι → ι . x0 x3x3 x2 = x1 x2
Let x3 of type ιι be given.
Assume H3: x0 x3.
Apply H1 with x1, x3, λ x4 x5 : ι → ι . x3 x2 = x5 x2 leaving 3 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Let x4 of type ιιο be given.
Assume H4: x4 (x3 x2) (x3 x2).
The subproof is completed by applying H4.
Claim L4: ∀ x3 : ι → ι . x0 x3x3 x2 = Descr_ii x0 x2
Apply unknownprop_c3f0de4cb966012957ca752938aa96a32c594389e7aea45227d571c0506618ba with λ x3 . ∀ x4 : ι → ι . x0 x4x4 x2 = x3, x1 x2.
The subproof is completed by applying L3.
Apply L4 with x1.
The subproof is completed by applying H2.
Apply L3 with λ x2 x3 : ι → ι . x0 x2.
The subproof is completed by applying H2.