
theorem
for L being well-unital add-associative right_zeroed right_complementable
            associative Abelian distributive non degenerated doubleLoopStr
for p being Polynomial of L
holds deg even_part(p) <= deg p & deg odd_part(p) <= deg p
proof
let L be add-associative right_zeroed right_complementable associative
         Abelian well-unital distributive non degenerated doubleLoopStr;
let p be Polynomial of L;
set e = even_part(p), o = odd_part(p);
per cases;
suppose p = 0_.(L);
  hence thesis by Th7;
  end;
suppose p <> 0_.(L);
  then reconsider pp = p as non zero Polynomial of L by UPROOTS:def 5;
  p = e + o by Th9;
  then A1: deg pp = max(deg(e),deg(o)) by Th21,HURWITZ:21;
  hence deg even_part(p) <= deg p by XXREAL_0:25;
  thus deg odd_part(p) <= deg p by A1,XXREAL_0:25;
  end;
end;
