We complete the proof of "Feynman rules" for constructing M-point conformal
blocks with external and internal scalars in any topology for arbitrary M in
any spacetime dimension by combining the rules for the blocks (based on their
Witten diagram interpretation) with the rules for the construction of conformal
cross ratios (based on OPE flow diagrams). The full set of Feynman rules leads
to blocks as power series of the hypergeometric type in the conformal cross
ratios. We then provide a proof by recursion of the Feynman rules which relies
heavily on the first Barnes lemma and the decomposition of the topology of
interest in comb-like structures. Finally, we provide a nine-point example to
illustrate the rules.Comment: 61 pages + an appendi