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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιι be given.
Let x2 of type ιιο be given.
Let x3 of type ι be given.
Assume H0: x3Sep2 x0 x1 x2.
Apply SepE with lam x0 (λ x4 . x1 x4), λ x4 . x2 (ap x4 0) (ap x4 1), x3, ∃ x4 . and (x4x0) (∃ x5 . and (x5x1 x4) (and (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) (x2 x4 x5))) leaving 2 subgoals.
The subproof is completed by applying H0.
Assume H1: x3lam x0 (λ x4 . x1 x4).
Assume H2: x2 (ap x3 0) (ap x3 1).
Apply lamE2 with x0, x1, x3, ∃ x4 . and (x4x0) (∃ x5 . and (x5x1 x4) (and (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)) (x2 x4 x5))) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x4 of type ι be given.
Assume H3: (λ x5 . and (x5x0) (∃ x6 . and (x6x1 x5) (x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)))) x4.
Apply H3 with ∃ x5 . and (x5x0) (∃ x6 . and (x6x1 x5) (and (x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)) (x2 x5 x6))).
Assume H4: x4x0.
Assume H5: ∃ x5 . and (x5x1 x4) (x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5)).
Apply H5 with ∃ x5 . and (x5x0) (∃ x6 . and (x6x1 x5) (and (x3 = lam 2 (λ x7 . If_i (x7 = 0) x5 x6)) (x2 x5 x6))).
Let x5 of type ι be given.
Assume H6: (λ x6 . and (x6x1 x4) (x3 = lam 2 (λ x7 . If_i (x7 = 0) x4 x6))) x5.
Apply H6 with ∃ x6 . and (x6x0) (∃ x7 . and (x7x1 x6) (and (x3 = lam 2 (λ x8 . If_i (x8 = 0) x6 x7)) (x2 x6 x7))).
Assume H7: x5x1 x4.
Assume H8: x3 = lam 2 (λ x6 . If_i (x6 = 0) x4 x5).
Let x6 of type ο be given.
Assume H9: ∀ x7 . and (x7x0) (∃ x8 . and (x8x1 x7) (and (x3 = lam 2 (λ x9 . If_i (x9 = 0) ... ...)) ...))x6.
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