
theorem
  for L being well-unital non degenerated non empty doubleLoopStr for
z being Element of L for k being Element of NAT st k >= 1 holds deg qpoly(k,z)
  = k - 1
proof
  let L be well-unital non degenerated non empty doubleLoopStr;
  let z be Element of L;
  let k be Element of NAT;
  set p = qpoly(k,z);
A1: k - 1 < k - 0 by XREAL_1:10;
  assume k >= 1;
  then k - 1 >= 1 - 1 by XREAL_1:9;
  then reconsider k9 = k - 1 as Element of NAT by INT_1:3;
  k - k9 - 1 = 0;
  then p.(k-1) = power(L).(z,0) by A1,Def4
    .= 1_L by GROUP_1:def 7;
  then
A2: p.(k-1) <> 0.L;
A3: now
    let m be Nat;
    assume
A4: m is_at_least_length_of p;
    now
      assume k > m;
      then k9 + 1 > m;
      then k9 >= m by NAT_1:13;
      hence contradiction by A2,A4,ALGSEQ_1:def 2;
    end;
    hence k <= m;
  end;
  for i being Nat st i >= k holds p.i = 0.L by Def4;
  then k is_at_least_length_of p by ALGSEQ_1:def 2;
  hence thesis by A3,ALGSEQ_1:def 3;
end;
