Lattice of Substitutions is a Heyting Algebra

(6) For every finite element a of V→̇C holds {a} ∈ SubstitutionSet(V,C). (7) If A a B = A, then for every set a such that a ∈ A there exists a set b such that b ∈ B and b⊆ a. (8) If μ(A a B) = A, then for every set a such that a ∈ A there exists a set b such that b ∈ B and b⊆ a. (9) If for every set a such that a ∈ A there exists a set b such that b ∈ B and b ⊆ a, then μ(A a B) = A. Let V be a set, let C be a finite set, and let A be an element of Fin(V→̇C). The functor InvolvedA is defined by: (Def. 1) x ∈ InvolvedA iff there exists a finite function f such that f ∈ A and x ∈ dom f .