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 Th52:
  for n be Nat
  for L be non empty ZeroStr
  for p be Series of n+1,L holds vars(p removed_last) c= vars p\{n}
proof
  let n be Nat;
  let L be non empty ZeroStr;
  let p be Series of n+1,L;
  set r=p removed_last;
  let y be object;
  assume y in vars(p removed_last);
  then consider b be bag of n such that
A1:b in Support r & b.y <> 0 by Def5;
  set bn=b bag_extend 0;
  A2: Bags(n+1)=dom p by PARTFUN1:def 2;
  0.L<> r.b = p.bn & bn in Bags(n+1) by A1,Def6,POLYNOM1:def 4;
  then
A3: bn in Support p by POLYNOM1:def 3,A2;
A4: y in dom b =n by A1,FUNCT_1:def 2,PARTFUN1:def 2;
  then (bn|n).y = bn.y by FUNCT_1:49;
  then b.y = bn.y by HILBASIS:def 1;
  then
A5:y in vars p by A3,A1,Def5;
  not n in n;
  hence thesis by A4,A5,ZFMISC_1:56;
end;
