(1) and (2), it is enough to use a simple induction that shows and work for , and such is proved the Theorem of the Unique Homomorphic Extension. Example of a particular case We can use the theorem of unique homomorphic extension for calculating numeric expressions over whole numbers. First, we must define the following: where Be Be he inductive closure of under and be Be Then will be a function that calculates recursively the truth-value of a proposition, and in a way, will be an extension of the function that associates a truth-value to each atomic proposition, such that: (1) (2) (Negation) (AND Operator) (OR Operator) (IF-THEN Operator) References Theorems in analysis