When giving an inductive definition of a function , it is important to only use in the definition of if is a substructure of .

For example, the "function" given inductively by and is not well-defined. Indeed, try computing ; any choice of output satisfies the defining equations.

As a more insidious example, consider given by and . Here, it is perfectly reasonable to use in the definition of because is a substructure of , but is not ok, because is not necessarily a substructure of and we can only apply to substructures of in the definition of .