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 Th39:
  for X be set, S be ZeroStr
    for p be Series of X,S holds
        vars p = union{ support b where b is Element of Bags X:b in Support p}
proof
  let X be set, S be ZeroStr;
  let p be Series of X,S;
  set F={ support b where b is Element of Bags X:b in Support p};
  thus vars p c= union F
  proof
    let x;
    assume x in vars p;
    then consider b be bag of X such that
A1: b in Support p & b.x <> 0 by Def5;
    b in Bags X by PRE_POLY:def 12;
    then x in support b & support b in F by A1,PRE_POLY:def 7;
    hence thesis by TARSKI:def 4;
  end;
  let x;
  assume x in union F;
  then consider B be set such that
A2: x in B & B in F by TARSKI:def 4;
  consider b be  Element of Bags X such that
A3: B= support b & b in Support p by A2;
  b.x<>0 by A2,A3,PRE_POLY:def 7;
  hence thesis by A3,Def5;
end;
