Definition:Assembly (Mathematical Theory)

Definition
An assembly in a mathematical theory is a succession of signs written one after another, along with other delineating marks according to the specific nature of the theory under consideration.

Bourbaki Definition
The mathematical theory as defined in allows certain signs which are not letters to be joined in pairs by links, as follows:
 * $\overbrace {\tau A \Box}^{} A'$

Concatenation
Let $\mathbf A$ and $\mathbf B$ denote assemblies.

Then $\mathbf {AB}$ denotes the assembly formed from writing the assembly $\mathbf B$ immediately to the right of assembly $\mathbf A$.

Extracting the Variable
(or whatever this operation is called)

Let $\mathbf A$ denote an assembly and let $x$ be a letter.

Then the assembly $\map {\tau_x} {\mathbf A}$ is constructed as follows:
 * 1) Form the assembly $\tau \mathbf A$
 * 2) Link each occurrence of $x$ wherever it appears in $\mathbf A$ to the $\tau$ written to the left of $\mathbf A$
 * 3) Replace $x$ by $\Box$ wherever it occurs in $\mathbf A$.

The assembly denoted $\map {\tau_x} {\mathbf A}$ therefore does not contain $x$.

Replacement
Let $\mathbf A$ and $\mathbf B$ denote assemblies.

We denote by:
 * $\paren {\mathbf B | x} \mathbf A$

the assembly obtained by replacing $x$, wherever it occurs, by $\mathbf B$.

This is read::
 * $\mathbf B$ replaces $x$ in $\mathbf A$.

If $x$ does not appear in $\mathbf A$, then $\paren {\mathbf B | x} \mathbf A$ is identical with $\mathbf A$.

In particular:
 * $\paren {\mathbf B | x} \map {\tau_x} {\mathbf A}$

is identical with $\map {\tau_x} {\mathbf A}$.