Hi,
One has to take in mind that the above is true only if the lens image circle is "just enough" as to cover the above mentioned image areas... The circle on confusion value (d) increases further with the image circle size... Therefore, there cases where one may try two different 180mm lenses on a (say) 4x5 image area and find that DOF of one is narrower that the other... The larger image circle lens will always have the narrower DOF.
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