On the Composition of Macro Instructions of Standard Computers

We follow the rules: k, m are natural numbers, x, X are sets, and N is a set with non empty elements. Let f be a function and let g be a non empty function. One can verify that f+·g is non empty and g+·f is non empty. Let f , g be finite functions. Note that f+·g is finite. Next we state two propositions: (1) For all functions f , g holds dom f ≈ dom g iff f ≈ g. (2) For all finite functions f , g such that dom f ∩ dom g = ∅ holds card(f+·g) = card f + card g. Let f be a function and let A be a set. Note that f \ A is function-like and relation-like. One can prove the following two propositions: (3) For all functions f , g such that x ∈ dom f \dom g holds (f \g)(x) = f(x). (4) For every non empty finite set F holds cardF − 1 = cardF − 1.