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 Th81:
  for X be set, S be add-associative right_zeroed right_complementable
    non empty addLoopStr
  for p,q be Series of X, S
  for V be set st vars p c= V & vars q c= V holds vars (p-q) c= V
proof
  let X be set, S be add-associative right_zeroed right_complementable
  non empty addLoopStr;
  let p,q be Series of X, S;
  let V be set;
  assume
A1: vars p c= V & vars q c= V;
  vars (-q) = vars (q) by Th42;
  then vars (p+ -q) c= vars p \/ vars (-q) c= V by A1,Th41,XBOOLE_1:8;
  then vars (p+ -q) c= V;
  hence thesis by POLYNOM1:def 7;
end;
