On Bag of 1. Part I

Summary The article concerns about formalizing multivariable formal power series and polynomials [3] in one variable terms of “bag” (as described detail [9]), the same notion as multiset over a finite set, Mizar system [1], [2]. Polynomial rings ring series, both variable, have been formalized [6], [5] respectively, elements these are represented by infinite sequences scalars. On other hand, formalization multivariate polynomial requires extra techniques using to represent monomials variables, function from bags variables scalar ring. This means way construction different between single multi case (which implies some tedious constructions, e.g. ten [8], or generally problem prime representing [7]). Introducing bag-based provides straight apply mathematical induction with respect number variables. Another consequence article, is subring an algebra [4] ring, namely corresponding series. A sketch actual consists following four steps: 1. translation Bags 1 (the set all singleton) N; 2. commutative denoted Formal-Series ( n, R ); 3. restricting Polynom-Ring (1, ). proof fact that ), -Algebra, included well; 4. isomorphism existing given sequence: ) →˜ .