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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: ∀ x1 . In x1 x0and (setsum_p x1) (In (ap x1 0) 2).
Claim L1: setsum (ReplSep x0 (λ x1 . In (setsum 0 (ap x1 1)) x0) (λ x1 . ap x1 1)) (ReplSep x0 (λ x1 . In (setsum 1 (ap x1 1)) x0) (λ x1 . ap x1 1)) = x0
Apply unknownprop_219a5692ece616b4a88502d80a85b644180cde982b21251f92a23d11d1a5d022 with setsum (ReplSep x0 (λ x1 . In (setsum 0 (ap x1 1)) x0) (λ x1 . ap x1 1)) (ReplSep x0 (λ x1 . In (setsum 1 (ap x1 1)) x0) (λ x1 . ap x1 1)), x0 leaving 2 subgoals.
Let x1 of type ι be given.
Assume H1: In x1 (setsum (ReplSep x0 (λ x2 . In (setsum 0 (ap x2 1)) x0) (λ x2 . ap x2 1)) (ReplSep x0 (λ x2 . In (setsum 1 (ap x2 1)) x0) (λ x2 . ap x2 1))).
Apply unknownprop_976b9ed71c1ec1f277c9c37a01879b51c2de3497fe82149802bec54f853970e6 with ReplSep x0 (λ x2 . In (setsum 0 (ap x2 1)) x0) (λ x2 . ap x2 1), ReplSep x0 (λ x2 . In (setsum 1 (ap x2 1)) x0) (λ x2 . ap x2 1), x1, λ x2 . In x2 x0 leaving 3 subgoals.
The subproof is completed by applying H1.
Let x2 of type ι be given.
Assume H2: In x2 (ReplSep x0 (λ x3 . In (setsum 0 (ap x3 1)) x0) (λ x3 . ap x3 1)).
Apply unknownprop_021a576837934491f6aaf936d4c5a9c68d45f2b77fcd13cc395cfdeec72f7dac with x0, λ x3 . In (setsum 0 (ap x3 1)) x0, λ x3 . ap x3 1, x2, In (setsum 0 x2) x0 leaving 2 subgoals.
The subproof is completed by applying H2.
Let x3 of type ι be given.
Assume H3: In x3 x0.
Assume H4: In (setsum 0 (ap x3 1)) x0.
Assume H5: x2 = ap x3 1.
Apply H5 with λ x4 x5 . In (setsum 0 x5) x0.
The subproof is completed by applying H4.
Let x2 of type ι be given.
Assume H2: In x2 (ReplSep x0 (λ x3 . In (setsum 1 (ap x3 1)) x0) (λ x3 . ap x3 1)).
Apply unknownprop_021a576837934491f6aaf936d4c5a9c68d45f2b77fcd13cc395cfdeec72f7dac with x0, λ x3 . In (setsum 1 ...) ..., ..., ..., ... leaving 2 subgoals.
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Apply L1 with λ x1 x2 . setsum_p x1.
The subproof is completed by applying unknownprop_f61ccefc6bc57eb6c116b3bc3f27a552fe11c91770c4e9cfa989285bab91c3f5 with ReplSep x0 (λ x1 . In (setsum 0 (ap x1 1)) x0) (λ x1 . ap x1 1), ReplSep x0 (λ x1 . In (setsum 1 (ap x1 1)) x0) (λ x1 . ap x1 1).