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Suppose x and y are two rational numbers. Therefore x = p/q and y = r/s where p, q, r and s are integers and q and s are not zero.

Then x - y = p/q - r/s = ps/qs - qr/qs = (ps - qr)/qs

By the closure of the set of integers under multiplication, ps, qr and qs are all integers,

by the closure of the set of integers under subtraction, (ps - qr) is an integer,

and by the multiplicative properties of 0, qs is non zero.

Therefore (ps - qr)/qs satisfies the requirements of a rational number.


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