Intuitively there is a drastic distinction between the pure decentralized block-chain systems like Defis and those that only utilize block-chain as an enhancing technology but remain centralized with real-world business model and conventional technologies like database, application server, etc. Our study explores extensively this distinction from a methodological point of view, classifies them into blockchain-complete and blockchain-partial, analyzes key features of the two types, and reveals the root cause of this distinction. We analyze the function or, in more strong words, the "ultimate purpose" of blockchain in the blockchain-partial systems, and present a conceptual model we named proof-chain that quite satisfactorily represented the general paradigm of blockchain in blockchain-partial systems. A universal tension between strength of proof-chain and privacy is then revealed and the zero-knowledge based proof-chain takes shape. Several case studies demonstrate the explaining power of our proof-chain methodology. We then apply proof-chain methodology to the analysis of the ecosystem of a collaborating group of blockchain-partial systems, representing the paradigm of public and private data domain whose border the proof-chain crosses. Finally, some derived guidelines from this methodology speak usefulness of our methodology.