Construction of Gröbner bases. S-Polynomials and Standard Representations

One can prove the following propositions: (1) For every set X and for every finite sequence p of elements of X such that p 6= ∅ holds p↾1 = 〈p1〉. (2) Let L be a non empty loop structure, p be a finite sequence of elements of L, and n, m be natural numbers. If m ¬ n, then p↾n↾m = p↾m. (3) Let L be an add-associative right zeroed right complementable non empty loop structure, p be a finite sequence of elements of L, and n be a natural number. Suppose that for every natural number k such that k ∈ dom p and k > n holds p(k) = 0L. Then ∑ p = ∑ (p↾n). (4) Let L be an add-associative right zeroed Abelian non empty loop structure, f be a finite sequence of elements of L, and i, j be natural numbers. Then ∑ Swap(f, i, j) = ∑ f.