Does the existence of an indirect proof imply the existence of a direct proof?

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So direct proofs are often considered more informative than indirect ones. This got me thinking, and as far as I believe this question has no practical implication - does the existence of an indirect proof imply the existence of a direct proof, for any given problem in any given field?
I know this might just be a yes or no answer, but is there an explanation for it that even a layman can understand?