I'm reading Calculus on Manifold by Spivak and I am beffudled by a particular derivation on page 35 in the proof of the Inverse Function Theorem.
Let $\lambda$ be the linear transformation $Df(a)$. Then $\lambda$ is non-singular, since $\det f'(a) \ne 0$. Now
$$D(\lambda^{-1} \circ f)(a)$$ $$=D(\lambda^{-1})(f(a)) \circ Df(a)$$ $$=\lambda^{-1} \circ Df(a)$$ $$=I$$
I can see how the first and second lines are equivalent. And it's obvious how the third line is the identity linear transformation. But how does one go from the second to the third line? It seems to imply that $D(\lambda^{-1})(f(a))=\lambda^{-1}$ which seems very non-obvious to me.