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 Th84:
  for X be Ordinal
  for L be non empty ZeroStr
  for s be Series of X,L
  for perm be Permutation of X
    holds vars (s permuted_by perm) c= perm.:vars(s)
proof
  let X be Ordinal;
  let L be non empty ZeroStr;
  let s be Series of X,L;
  let perm be Permutation of X;
  set S=s permuted_by perm;
  let x;
  assume x in vars S;
  then consider b be bag of X such that
A1: b in Support S & b.x <> 0 by Def5;
A2: b*perm in Support s by A1,HILB10_2:21;
  reconsider B=b*perm as bag of X;
A3: x in dom b =X by A1,FUNCT_1:def 2,PARTFUN1:def 2;
  rng perm =X=dom perm by FUNCT_2:def 3,52;
  then consider y be object such that
A4: y in dom perm & perm.y =x by A3,FUNCT_1:def 3;
  y in dom (b*perm) by A4,A3, FUNCT_1:11;
  then B.y <>0 by A1,A4,FUNCT_1:12;
  then y in vars(s) by A2,Def5;
  hence thesis by A4,FUNCT_1:def 6;
end;
