"I think the Church is that body that is descended from the body that Jesus started:"
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As I am sure you are aware, one of the never ending discussions here is about the primacy, or lack thereof, of the Roman Catholic Church. Are you ascribing to the view that because they claim succession through Peter that they are "The CHURCH"?
The RCC has a traceable lineage back to the early church. They aren't the only ones who can, though. And then there's always the issue of spiritual lineage to keep excesses in check.