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Cyan comments on Looking for proof of conditional probability - Less Wrong
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...which is done explicitly in Jay Kadane's free text,starting on page 29.
Snapping pointers: direct link. Commercial use forbidden (probably the reason for the pointer chain).
Ah, thanks. (It's done semi-explicitly right on the wiki page, though. Or at least an effectively general example is set up and the form of the proof is described. (ie, the only way that there wouldn't automatically be a solution to the equations to dutch book you would be if the system had a determinant of zero, and doing so forces the standard rule for conditional probability))