
theorem
  for L being add-associative right_zeroed right_complementable non
empty addLoopStr for p1,p2 being Polynomial of L st deg p1 <> deg p2 holds deg
  (p1 + p2) = max(deg(p1),deg(p2))
proof
  let L be add-associative right_zeroed right_complementable non empty
  addLoopStr;
  let p1,p2 be Polynomial of L;
  assume deg p1 <> deg p2;
  then
A1: deg(p1+p2) = max(len(p1),len(p2)) - 1 by POLYNOM4:7;
  per cases by XXREAL_0:16;
  suppose
A2: max(len(p1),len(p2)) = len(p1);
    then len p2 <= len p1 by XXREAL_0:25;
    then deg p2 <= deg p1 by XREAL_1:9;
    hence thesis by A1,A2,XXREAL_0:def 10;
  end;
  suppose
A3: max(len(p1),len(p2)) = len(p2);
    then len p1 <= len p2 by XXREAL_0:25;
    then deg p1 <= deg p2 by XREAL_1:9;
    hence thesis by A1,A3,XXREAL_0:def 10;
  end;
end;
