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Let ##(f,g) \in B^A## where ##A## and ##B## are non-empty sets, ##B^A## denotes the set of bijective functions between ##A## and ##B##.

We assume that there exists ##h_0: A \rightarrow A## and ##h_1: B \rightarrow B## such that ##f = h_1 \circ g \circ h_0 ##.

This implies that ##g = h^{-1}_1 \circ f \circ h^{-1}_0##, according to my teacher, but why is that?

We have that ##h^{-1}_1 \circ f = g \circ h^{-1}_0 ##, but how do I proceed from that?

I have drawn a graph with sets and arrows representing functions such that going from ##A## to ##B## via ##h_0##, ##g## and ##h_1## is the same as going from ##A## to ##B## via ##f##. I then manipulated this graph a little while maintaining the proper relations between the sets, which showed that going from ##A## to ##B## via ##g## is the same as going through (in order) ##h^{-1}_0##, ##f## and ##h^{-1}_1##, but this is hardly a proof at all.

Thanks

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# Invert a triple composite function p(q(r(x)))

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