reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
          right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;
reserve O for Ordinal,
        R for right_zeroed add-associative right_complementable
             right_unital distributive non trivial doubleLoopStr,
        p for Polynomial of O, R;

theorem Th44:
  for X be set, S be add-associative right_zeroed right_complementable
    right-distributive non empty doubleLoopStr
  for p be Series of X,S, a be Element of S
    holds vars (a * p) c= vars p
proof
  let X be set, S be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr;
  let p be Series of X,S, a be Element of S;
  let x;
  assume x in vars (a*p);
  then consider b be bag of X such that
A1: b in Support (a*p) & b.x <> 0 by Def5;
  a * (p.b) = (a*p).b <>0.S by A1,POLYNOM1:def 3,POLYNOM7:def 9;
  then p.b <>0.S & b in Bags X = dom p
  by PRE_POLY:def 12,PARTFUN1:def 2;
  then b in Support p by POLYNOM1:def 3;
  hence thesis by A1,Def5;
end;
