The first step consists of standardizing the projections making up the composite. An equivalence on the image of a projection, when composed with its kernel, yields the kernel of its composite with any map having the equivalence as kernel: composites of projections can thus be presented as a projection followed by bijections between their images. Let J be the image of the first projection in this presentation, thus with the same kernel as the composite f, hence can be described via a bijection of f 's image I, which sends each element i to some preimage in f y1 i.
Assuming this first projection given, when can it be followed by two Ε½ . further projections to yield f ? The third projection must fix since onto I.
Ε½ . Hence j g J l I, if unequal to the composite f j , must be moved by the Ε½ . second projection: if to I, then only to the composite f j , fixed by the Ε½ . Ε½ . third projection. If also f j g J l I _ F, the fixpoints of f, it must also be moved, so j must go to D, the complement of I; moreover, to no jΠ g D, since the second projection is injective on J. The basic requirement is thus that D _ J be at least as numerous as the graph G of f in Ε½ . J l I minus the fixpoints F: i.e., the pairs j, f j g J l I _ F. Indeed, the second projection could then send these j bijectively into D _ J, send the Ε½ . remaining j g J l I to their images f j , and fix these and D, while the third would send the j injected in D to their image in I, keeping I fixed.
At last, we determine when there exists this kind of a set J of representatives for the classes of Ker f ; it would suffice to start with an arbitrary J and improve it until its complement in D is at least as numerous as its G.
Ε½ . If a class includes a fixpoint there can be at most one , choose it; if not, Ε½ and it includes an element of I all of whose preimages are in D i.e., of