
theorem Th22:
  for L being add-associative right_zeroed right_complementable
  Abelian non empty addLoopStr for p1,p2 being Polynomial of L holds deg(p1 +
  p2) <= max(deg(p1),deg(p2))
proof
  let L be add-associative right_zeroed right_complementable Abelian non
  empty addLoopStr;
  let p1,p2 be Polynomial of L;
  per cases;
  suppose
A1: p1 = 0_.(L);
    then deg(p1) = -1 by Th20;
    then
A2: deg(p2) >= deg(p1) by Lm9;
    deg(p1 + p2) = deg(p2) by A1,POLYNOM3:28
      .= max(deg(p1),deg(p2)) by A2,XXREAL_0:def 10;
    hence thesis;
  end;
  suppose
A3: p2 = 0_.(L);
    then deg(p2) = -1 by Th20;
    then
A4: deg(p1) >= deg(p2) by Lm9;
    deg(p1 + p2) = deg(p1) by A3,POLYNOM3:28
      .= max(deg(p1),deg(p2)) by A4,XXREAL_0:def 10;
    hence thesis;
  end;
  suppose
A5: p1 <> 0_.(L) & p2 <> 0_.(L);
    then
A6: deg(p2) is Element of NAT by Lm8;
    deg(p1) is Element of NAT by A5,Lm8;
    then reconsider m = max(deg(p1),deg(p2)) as Element of NAT by A6,
XXREAL_0:16;
    for k being Nat st k >= m+1 holds (p1+p2).k = 0.L
    proof
      let k be Nat;
      assume
A7:   k >= m + 1;
      deg(p2) <= m by XXREAL_0:25;
      then deg(p2) + 1 <= m + 1 by XREAL_1:6;
      then
A8:   p2.k = 0.L by A7,ALGSEQ_1:8,XXREAL_0:2;
      deg(p1) <= m by XXREAL_0:25;
      then deg(p1) + 1 <= m + 1 by XREAL_1:6;
      then p1.k = 0.L by A7,ALGSEQ_1:8,XXREAL_0:2;
      hence (p1+p2).k = 0.L + 0.L by A8,NORMSP_1:def 2
        .= 0.L by RLVECT_1:def 4;
    end;
    then m+1 is_at_least_length_of (p1+p2) by ALGSEQ_1:def 2;
    then len(p1+p2)<=m+1 by ALGSEQ_1:def 3;
    then len(p1+p2)-1<=m+1-1 by XREAL_1:9;
    hence thesis;
  end;
end;
