The unique homomorphic extension theorem is a result in mathematical logic which formalizes the intuition that the truth or falsity of a statement can be deduced from the truth values of its parts. The lemma Let A be a non-empty set, X a subset of A, F a set of functions in A, and the inductive closure of X under F. Let be B any non-empty set and let G be the set of functions on B, such that there is a function in G that maps with each function f of arity n in F the following function in G (G cannot be a bijection). From this lemma we can now build the concept of unique homomorphic extension. The theorem If is a free set generated by X and F, for each function there is a single function such that: For each function f of arity n > 0, for each Consequence The identities seen in (1) e (2) show that is an homomorphism, specifically named the unique homomorphic extension of . To prove the theorem, two requirements must be met: to prove that the extension () exists and is unique (assuring the lack of bijections). Proof of the theorem We must define a sequence of functions inductively, satisfying conditions (1) and (2) restricted to . For this, we define , and given then shall have the following graph: First we must be certain the graph actually has functionality, since is a free set, from the lemma we have when , so we only have to determine the functionality for the left side of the union. Knowing that the elements of G are functions(again, as defined by the lemma), the only instance where and for some is possible is if we have for some and for some generators and in . Since and are disjoint when this implies and . Being all in , we must have . Then we have with , displaying functionality. Before moving further we must make use of a new lemma that determines the rules for partial functions, it may be written as: (3)Be a sequence of partial functions such that . Then, is a partial function. Using (3), is a partial function. Since then is total in . Furthermore, it is clear from the definition of that satisfies (1) and (2). To prove the uniqueness of , or any other function that satisfies