The part to the right of add in the Implementation add part of the definition is called a capsule. The purpose of a capsule is:
What is a local environment? First, what is an environment? environment Think of the capsule as an input file that Axiom reads from top to bottom. Think of the input file as having a )clear all at the top so that initially no variables or functions are defined. When this file is read, variables such as and in DrawComplex are set to initial values. Also, all the functions defined in the capsule are compiled. These include those that are exported (like ), and those that are not (like ). At the end, you get a set of name-value pairs: variable names (like and ) are paired with assigned values, while operation names (like and ) are paired with function values.
This set of name-value pairs is called an environment. Actually, we call this environment the ``initial environment'' of a package: it is the environment that exists immediately after the package is first built. Afterwards, functions of this capsule can access or reset a variable in the environment. The environment is called local since any changes to the value of a variable in this environment can be seen only by these functions.
Only the functions from the package can change the variables in the local environment. When two functions are called successively from a package, any changes caused by the first function called are seen by the second.
Since the environment is local to the package, its names don't get mixed up with others in the system or your workspace. If you happen to have a variable called in your workspace, it does not affect what the DrawComplex functions do in any way.
The functions in a package are compiled into machine code. Unlike function definitions in input files that may be compiled repeatedly as you use them with varying argument types, functions in packages have a unique type (generally parameterized by the argument parameters of a package) and a unique compilation residing on disk.
The capsule itself is turned into a compiled function. This so-called capsule function is what builds the initial environment spoken of above. If the package has arguments (see below), then each call to the package constructor with a distinct pair of arguments builds a distinct package, each with its own local environment.