Book IX · Proposition 26

IX.26

If from an odd number an odd number be subtracted, the remainder will be even.Heath, 1908

Every step, checked

• an odd number has an odd subtractedgiven
• the subtraction is a proper onegiven
• the remainder is evenby IX.22

What it needs, and what needs it

Needs: IX.22

Rests on: Def.VII.6

Depth: 2 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.26", THEOREM,
             sample=lambda rng: _pair_by_parity(rng, False, False))
def prop_IX_26(a: int, b: int) -> Out:
    hypothesis("an odd number has an odd subtracted",
               not measures(2, a) and not measures(2, b))
    hypothesis("the subtraction is a proper one", a > b, guard=True)
    because(prop_IX_22, [a, b]) if all(not measures(2, n) for n in (a, b)) else None

    claim("the remainder is even", "IX.22", measures(2, a - b))
    return Out(remainder=a - b)