If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them.
Proof
With a=mb, c=md, and subtractions a′=nb, c′=nd:
a−a′=(m−n)b and c−c′=(m−n)d by Common Notion 3. If m=n
the remainders are zero (equal); otherwise both remainders are
(m−n)-fold of b, d respectively.