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 Th41:
  for X be set, S be right_zeroed non empty addLoopStr
  for p,q be Series of X, S holds
    vars (p+q) c= vars p \/ vars q
proof
  let X be set, S be right_zeroed non empty addLoopStr;
  let p,q be Series of X, S;
  let x;
  assume x in vars (p+q);
  then consider b be bag of X such that
A1: b in Support (p+q) & b.x <> 0 by Def5;
  Support (p+q) c= (Support p) \/(Support q) by POLYNOM1:20;
  then b in (Support p) or b in (Support q) by A1,XBOOLE_0:def 3;
  then x in vars p or x in vars q by A1,Def5;
  hence thesis by XBOOLE_0:def 3;
end;
