Technical Preliminaries to Algebraic Specifications

One can prove the following propositions: (1) For all functions f , g, h such that dom f ∩ dom g ⊆ domh holds f+·g+·h = g+·f+·h. (2) For all functions f , g, h such that f ⊆ g and rng h ∩ dom g ⊆ dom f holds g · h = f · h. (3) For all functions f , g, h such that dom f ⊆ rng g and dom f misses rng h and g domh misses dom f holds f · (g+·h) = f · g. (4) For all functions f1, f2, g1, g2 such that f1 ≈ f2 and g1 ≈ g2 holds f1 · g1 ≈ f2 · g2. (5) Let X1, Y1, X2, Y2 be non empty sets, f be a function from X1 into X2, and g be a function from Y1 into Y2. If f ⊆ g, then f ∗ ⊆ g. (6) Let X1, Y1, X2, Y2 be non empty sets, f be a function from X1 into X2, and g be a function from Y1 into Y2. If f ≈ g, then f ∗ ≈ g. Let X be a set and let f be a function. The functor X -indexing f yielding a many sorted set indexed by X is defined as follows: (Def. 1) X -indexing f = idX+·f↾X. We now state a number of propositions: