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Your first assumption, that "Nothing can exist without purpose. Anything that exists must have to exist, or it wouldnt" is a mere statement of faith. "purpose" is an anthromorphic construct and has no absolute meaning. In order for your proof to be conclusive, assumption # 1 would not only have to be universally agreed upon but true! And while the statement: "Nothing can exist without purpose" can be argued, it cannot be proven.
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