
theorem Th40:
  for L being non degenerated well-unital non empty doubleLoopStr
  , n being non zero Element of NAT holds len unital_poly(L,n) = n+1
proof
  let L be non degenerated well-unital non empty doubleLoopStr;
  let n be non zero Element of NAT;
A1: for m being Nat st m is_at_least_length_of unital_poly(L,n) holds n+1 <= m
  proof
    let m be Nat such that
A2: m is_at_least_length_of unital_poly(L,n);
    now
      assume m < n+1;
      then m <= n by NAT_1:13;
      then unital_poly(L,n).(n) = 0.L by A2,ALGSEQ_1:def 2;
      hence contradiction by Th38;
    end;
    hence thesis;
  end;
A3: n+1 in NAT by ORDINAL1:def 12;
  for i being Nat st i >= n+1 holds unital_poly(L,n).i=0.L
  proof
    let i be Nat such that
A4: i >= n+1;
    now
A5:   n + 0 < n + 1 by XREAL_1:8;
      assume i = n;
      hence contradiction by A4,A5;
    end;
    hence thesis by A4,Th39;
  end;
  then n+1 is_at_least_length_of unital_poly(L,n) by ALGSEQ_1:def 2;
  hence thesis by A1,ALGSEQ_1:def 3,A3;
end;
