G.367 number continually doubled

IX.34

If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both even-times even and even-times odd.

number_continually_doubled()

prove A is both even-times even and even-times odd

assert Let the number A neither be one of those doubled from a dyad, nor have its half odd

∵ even_times_even_defassert Now that A is even-times even is manifest; for it has not its half odd

assert I say next that it is also even-times odd

assert If we bisect A, then bisect its half, and do this continually, we shall come upon some odd number which will measure A according to an even number

assert If not, we shall come upon a dyad, and A will be among those which are doubled from a dyad: which is contrary to the hypothesis

assert A is even-times odd

assert It was also proved even-times even

∴ A is both even-times even and even-times odd

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