Book IX · Proposition 33

IX.33

If a number have its half odd, it is even-times odd only.Heath, 1908

Every step, checked

• the half is oddgiven
• the number is even-times odd onlyby Def.VII.9

What it needs, and what needs it

Needs: nothing earlier.

Rests on: Def.VII.9

Depth: 0 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition("IX.33", THEOREM,
             sample=lambda rng: (2 * rng.randint(1, 20) + 1,))
def prop_IX_33(half: int) -> Out:
    number = 2 * half
    hypothesis("the half is odd", not measures(2, half))
    claim("the number is even-times odd only", "Def.VII.9",
          measures(2, number) and not measures(4, number))
    return Out(number=number)