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 Th42:
  for X be set, S be add-associative right_zeroed right_complementable
     non empty addLoopStr
  for p be Series of X, S holds
    vars p = vars -p
proof
  let X be set, S be add-associative right_zeroed right_complementable
    non empty addLoopStr;
  let p be Series of X, S;
  thus vars p c= vars -p
  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;
    reconsider b as Element of Bags X by PRE_POLY:def 12;
    -(p.b) <> - 0.S = 0.S by A1,POLYNOM1:def 4;
    then (-p).b <>0.S by POLYNOM1:17;
    then b in Support (-p) by POLYNOM1:def 4;
    hence thesis by A1,Def5;
  end;
  let x;
  assume x in vars (-p);
  then consider b be bag of X such that
A2: b in Support (-p) & b.x <> 0 by Def5;
  reconsider b as Element of Bags X by PRE_POLY:def 12;
  -((-p).b) <> - 0.S = 0.S by A2,POLYNOM1:def 4;
  then (- - p).b <>0.S by POLYNOM1:17;
  then b in Support (- -p) by POLYNOM1:def 4;
  then b in Support (p) by POLYNOM1:19;
  hence thesis by A2,Def5;
end;
