G.282 numbers prime sum¶
If two numbers be prime to one another, the sum is prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers is prime to one another.
numbers_prime_sum()
prove the sum / A C / is also prime to each of the numbers / A B /, / B C /; / A B /, / B C / are also prime to one another
construct Let two numbers / A B /, / B C / prime to one another be added
assert Let a number measure them, and let it be D
assert then D measures / A C /, / A B /,
assert it will also measure / B C /
assert It also measures / A B /
assert no number will measure the numbers / A C /, / A B / ; therefore / A C /, / A B / are prime to one another
assert For the same reason
assert / A C /, / B C / are also prime to one another
assert / A C / is prime to each of the numbers / A B /, / B C /
assert Let / A C /, / A B / be prime to one another
assert If / A B /, / B C / are not prime to one another, some number will measure / A B /, / B C /
assert Let a number measure them, and let it be D
assert D measures each of the numbers / A B /, / B C /,
assert It will also measure / A C /
assert It also measures / A B /
assert no number will measure the numbers / A B /, / B C /
∴ / A B /, / B C / are prime to one another