Let A and B be two sets. Let A₁, A₂ and A₃ be subsets of A such that A₁ A₂ A₃=A and A_i A_j= for i j . Let f: A B be a function. For each i=1,2,3 , let f_i: A_i B be defined by f_i(x)=f(x) for x A_i . Then

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1. Af is surjective whenever f_i is surjective for each i=1,2,3 .
2. Bf is injective whenever f_i is injective for each i=1,2,3 .
3. Cf is bijective whenever f_i is bijective for each i=1,2,3 .
4. Df_i is surjective for each i=1,2,3 whenever f is surjective.