Divisibility relation: transitivity proof

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Proof

I'm a bit confused about the proof for this relation. I get the first part, but the second line is where I'm totally muffled!

Any help is greatly appreciated :)

** An example would also be helpful

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You wish to show that $c\mid(ma+nb)$, in other words, there exists a $k$ such that $ma+nb=ck$. This is exactly what the second line does, it shows that $ma+nb=c(me+nf)$.