Theorem 3.1 (The Closure Property of Composition of Functions): If A, B, and C are sets and

and

are functions then the composition of f followed by g

is a function.

Proof: The rule in the definition does establish a relation between elements of A and elements of C. We must show that it is well defined. Assume

a = b

Then since f is a function, it follows from the well definedness of f , that

f(a) = f(b)

by Definition 3.2. Then since g is a function, it follows from the well definedness of g that

g(f(a)) = g(f(b))

and gf is well defined.