
theorem
  for L being non trivial ZeroStr, p being Polynomial of L st len p = 3
  ex a,b being Element of L, c being non zero Element of L st p = <%a,b,c%>
  proof
    let L be non trivial ZeroStr;
    let p be Polynomial of L;
    assume
A1: len p = 3;
    3 = 2+1;
    then p.2 <> 0.L by A1,ALGSEQ_1:10;
    then reconsider c = p.2 as non zero Element of L by STRUCT_0:def 12;
    take a = p.0, b = p.1, c;
    let n be Element of NAT;
    (n = 0 or ... or n = 2) or n > 2;
    then per cases;
    suppose n = 0 or n = 1 or n = 2;
      hence p.n = <%a,b,c%>.n by NIVEN:23,24,25;
    end;
    suppose n > 2;
      then
A2:   2+1 <= n by NAT_1:13;
      hence p.n = 0.L by A1,ALGSEQ_1:8
      .= <%a,b,c%>.n by A2,NIVEN:26;
    end;
  end;
