Let $f:A \to B, g:B \to C$. I don't really know how to prove this but I understand what it means.
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Then there exist $a,b$ with $a \neq b$ s.t. $g(a) = g(b)$. $f$ onto means there exist $c,d$ s.t. $f(c) = a, f(d) = b$. But then...
Does that help?