The Ring of Polynomials

Let D be a non empty set, let i be a natural number, and let p be a finite sequence of elements of D. Then p i is a finite sequence of elements of D. Let D be a non empty set and let a, b be elements of D. Then 〈a,b〉 is an element of D2. Let D be a non empty set, let k, n be natural numbers, let p be an element of Dk, and let q be an element of Dn. Then pa q is an element of Dk+n. Let D be a non empty set and let n be a natural number. Observe that every finite sequence of elements of Dn is finite sequence yielding. Let D be a non empty set, let k, n be natural numbers, let p be a finite sequence of elements of Dk, and let q be a finite sequence of elements of Dn. Then p _ q is an element of (Dk+n)∗. The scheme SeqOfSeqLambdaD deals with a non empty set A , a natural number B, a unary functor F yielding a natural number, and a binary functor G yielding an element of A , and states that: There exists a finite sequence p of elements of A∗ such that (i) len p = B, and (ii) for every natural number k such that k ∈ SegB holds len(pk) = F (k) and for every natural number n such that n ∈ dom(pk) holds pk(n) = G(k,n) for all values of the parameters.