The Evaluation of Multivariate Polynomials

In this article we present several logical schemes. The scheme FinRecExD2 deals with a non empty set A, an element B of A, a natural number C, and a ternary predicate P, and states that: There exists a finite sequence p of elements of A such that len p = C but p1 = B or C = 0 but for every natural number n such that 1 ¬ n and n < C holds P[n, pn, pn+1] provided the parameters meet the following conditions: • Let n be a natural number. Suppose 1 ¬ n and n < C. Let x be an element of A. Then there exists an element y of A such that P[n, x, y], and • Let n be a natural number. Suppose 1 ¬ n and n < C. Let x, y1, y2 be elements of A. If P[n, x, y1] and P[n, x, y2], then y1 = y2. The scheme FinRecUnD2 deals with a non empty set A, an element B of A, a natural number C, finite sequences D, E of elements of A, and a ternary predicate P, and states that: D = E provided the parameters meet the following requirements: • Let n be a natural number. Suppose 1 ¬ n and n < C. Let x, y1, y2 be elements of A. If P[n, x, y1] and P[n, x, y2], then y1 = y2,