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 x = p / q
y = r / s

x > y  =>  p  / q  > r  / s
let them be written such that q and s are positive.  p and r can be positive or negative.

Multiply with   q s   on both sides
=>     p s >  r q

p /q - r / s  =    [ p s - r q ] / q s
                 > 0  as  ps > rq and  qs  > 0

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