Book IX · Proposition 22

IX.22

If as many odd numbers as we please be added together, and their multitude be even, the whole will be even.Heath, 1908

Every step, checked

• all the numbers are oddgiven
• they are even in multitudegiven
• the sum is evenby IX.21

What it needs, and what needs it

Needs: IX.21

Used by: IX.23 IX.26

Rests on: Def.VII.6

Depth: 1 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.22",
    THEOREM,
    sample=lambda rng: _odd_numbers(rng, even_count=True),
)
def prop_IX_22(given: list) -> Out:
    hypothesis("all the numbers are odd", all(not measures(2, n) for n in given) and given)
    hypothesis("they are even in multitude", measures(2, len(given)))
    because(prop_IX_21, [2 * n for n in given])

    claim("the sum is even", "IX.21", measures(2, sum(given)))
    return Out(total=sum(given))