Terms as sections Theorem 4.3.9 substantiates the remarks about simultaneous substitution which we made after we first introduced the Substitution Lemma 1.1.5 . There may be more variables in the source context than are mentioned in the substituted terms, and it is not clear whether they should survive or be forgotten in the target. Indeed we took advantage of the ambiguity by saying that the substitutions for different variables commute . We use German and italic letters for the cables and single wires respectively. In the foregoing account we have chosen to colour the wires with variable-names.
Many authors prefer to number the pins in the plugs and sockets instead.
This ubiquitous renaming needlessly complicates the formal discussion of the notation. It is useful as an informal way of naming a single map, to avoid choosing names for the target variables. We have chosen this convention in order to take best advantage of variables as they are normally used in mathematics, namely to relate quantities defined in one part of any argument to their use in another. A change of free variables we call open a -equivalence . When we come to l -abstraction, terms differing only by the names of corresponding bound variables will be considered to be the same.