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 Th80:
  for X be set, S be add-associative right_zeroed right_complementable
    right-distributive non empty doubleLoopStr
  for p be Series of X,S, a be Element of S
  for V be set st vars p c= V holds vars (a * p) c= V
proof
  let X be set, S be add-associative right_zeroed right_complementable
  right-distributive non empty doubleLoopStr;
  let p be Series of X,S, a be Element of S;
  let V be set; assume A1: vars p c= V;
  vars (a * p) c= vars p by Th44;
  hence thesis by A1;
end;
