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Theorem
The operation of multiplication on the set of rational numbers∀x,y∈Q:x×y∈Q
Proof 1
Follows directly from the definition of rational numbers as the quotient field of the integral domainSo
Proof 2
From the definition of rational numbers, there exists four integersq≠0 s≠0 p q =x r s =y
p×r∈Z q×s∈Z
Therefore, by the definition of rational numbers:
x×y=p×r q×s ∈Q .
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