Let X, Tx, U and P be given.

Assume HP: partition_of_unity_family X Tx U P.

Apply HP to the current goal.

Assume Hleft _.

Apply Hleft to the current goal.

Assume Hleft2 _.

Apply Hleft2 to the current goal.

Assume Hleft3 _.

Apply Hleft3 to the current goal.

Assume _ HC.

An exact proof term for the current goal is HC.

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