Book IX · Proposition 25

IX.25

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

Every step, checked

• an even number has an odd subtractedgiven
• the subtraction is a proper onegiven
• the remainder is oddby IX.23

What it needs, and what needs it

Needs: IX.23

Used by: IX.27

Rests on: Def.VII.6

Depth: 3 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.25", THEOREM,
             sample=lambda rng: _pair_by_parity(rng, True, False))
def prop_IX_25(a: int, b: int) -> Out:
    hypothesis("an even number has an odd subtracted",
               measures(2, a) and not measures(2, b))
    hypothesis("the subtraction is a proper one", a > b, guard=True)
    because(prop_IX_23, [b]) if not measures(2, b) else None

    claim("the remainder is odd", "IX.23", not measures(2, a - b))
    return Out(remainder=a - b)