Two unequal numbers being set out, and the less being continually subtracted in turn from the greater, if the number which is left never measures the one before it until an unit is left, the original numbers is prime to one another.
unequal_numbers_set()
prove/ A B /, / C D / are prime to one another, that is, that an unit alone measures / A B /, / C D /
assert The less of two unequal numbers / A B /, / C D / being continually subtracted from the greater, let the number which is left never measure the one before it until an unit is left
assert If / A B /, / C D / are not prime to one another, some number will measure them
assert Let a number measure them, and let it be E
assert Let / C D /, measuring / F B /, leave / A F / less than itself, let / A F /, measuring / G D /, leave / C G / less than itself, and let / C G /, measuring / H F /, leave an unit / A H /
assert Since, then, E measures / C D /, and / C D / measures / F B /,
assertE also measures / F B /
assert It also measures / A B /
assert it will also measure / A F /
assert/ A F / measures / G D /
assertE also measures / G D /
assert It also measures / C D /
assert it will also measure / C G /
assert/ C G / measures / H F /
assertE also measures / H F /
assert It also measures / A F /
assert it will also measure the remainder, the unit / A H /, though it is a number ⇒⇐
∵numbers_prime_measured_def∴ no number will measure the numbers / A B /, / C D /; therefore / A B /, / C D / are prime to one another
