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;
