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In the end, the popes won supreme control of the church because

AOf their claim to be spiritual descendants of St. Peter
BOf religious differences with the patriarch of Constantinople
CBoth of their claim to be spiritual descendants of St. Peter and of their position in the original imperial capital of Rome
DOf their position in the original imperial capital of Rome

Verified Answer :

In the end, the popes won supreme control of the church because "c. Both of their claim to be spiritual descendants of St. Peter and of their position in the original imperial capital of Rome," since central authority at this time was crumbling. 

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