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 Th38:
  for X be Ordinal, S be non empty ZeroStr
    for p be Series of X,S holds
      vars p = {} iff p is Constant
proof
  let X be Ordinal, S be non empty ZeroStr;
  let p be Series of X,S;
  thus  vars p = {} implies p is Constant
  proof
    assume
A1: vars p = {};
    assume not p is Constant;
    then consider b be bag of X such that
A2: b <> EmptyBag X & p.b <> 0.S by POLYNOM7:def 7;
    b in Bags X = dom p by PARTFUN1:def 2,PRE_POLY:def 12;
    then
A3: b in Support p by A2, POLYNOM1:def 4;
    dom b = X = dom (EmptyBag X) by PARTFUN1:def 2;
    then ex x st
    x in X & (EmptyBag X).x<>b.x by A2;
    hence thesis by A1,A3,Def5;
  end;
  assume
A4:p is Constant;
  assume vars p <> {};
  then consider x such that
A5: x in vars p by XBOOLE_0:def 1;
  consider b be bag of X such that
A6: b in Support p & b.x <> 0 by A5,Def5;
  p.b <> 0.S & b <> EmptyBag X by A6,POLYNOM1:def 4;
  hence thesis by A4,POLYNOM7:def 7;
end;
