
theorem Th27:
  for L being well-unital non degenerated non empty doubleLoopStr
 for z being Element of L for k being Element of NAT holds deg rpoly(k,z) = k
proof
  let L be well-unital non degenerated non empty doubleLoopStr;
  let z be Element of L;
  let k be Element of NAT;
  set t = rpoly(k,z);
  set a = -power(L).(z,k);
  set f = (0,k)-->(a,1_L);
  per cases;
  suppose
A1: k = 0;
A2: now
      let m be Nat;
      assume
A3:   m is_at_least_length_of t;
      now
        assume m < 1;
        then
A4:     m = 0 by NAT_1:14;
A5:     k in {k} by TARSKI:def 1;
        then
A6:     k in dom({k}-->1_L);
        dom f = {0,k} by FUNCT_4:62;
        then 0 in dom f by TARSKI:def 2;
        then t.0 = f.0 by FUNCT_4:13
          .= ((0 .-->a)+*(k.-->1_L)).0 by FUNCT_4:def 4
          .= (0 .--> 1_L).0 by A1,A6,FUNCT_4:13
          .= 1_L by A1,A5,FUNCOP_1:7;
        hence contradiction by A3,A4,ALGSEQ_1:def 2;
      end;
      hence 1 <= m;
    end;
    now
      let i be Nat;
A7:   i in NAT by ORDINAL1:def 12;
      assume i >= 1;
      then not i in dom f by A1,TARSKI:def 2;
      hence t.i = (0_.(L)).i by FUNCT_4:11
        .= 0.L by A7,FUNCOP_1:7;
    end;
    then 1 is_at_least_length_of t by ALGSEQ_1:def 2;
    then len rpoly(k,z) = 1 by A2,ALGSEQ_1:def 3;
    hence thesis by A1;
  end;
  suppose
A8: k <> 0;
A9: now
      let m be Nat;
      assume
A10:  m is_at_least_length_of t;
      now
        assume m < k + 1;
        then
A11:    m <= k by NAT_1:13;
        t.k = 1_L by A8,Lm10;
        hence contradiction by A10,A11,ALGSEQ_1:def 2;
      end;
      hence k+1 <= m;
    end;
    now
      let i be Nat;
      assume i >= k+1;
      then i > k by NAT_1:13;
      hence t.i = 0.L by Lm11;
    end;
    then (k+1) is_at_least_length_of t by ALGSEQ_1:def 2;
    then len rpoly(k,z) = k + 1 by A9,ALGSEQ_1:def 3;
    hence thesis;
  end;
end;
