
theorem Th6:
  for L being non trivial ZeroStr, p being Polynomial of L st len p = 2
  ex a being Element of L, b being non zero Element of L st p = <%a,b%>
  proof
    let L be non trivial ZeroStr;
    let p be Polynomial of L;
    assume
A1: len p = 2;
    2 = 1+1;
    then p.1 <> 0.L by A1,ALGSEQ_1:10;
    then reconsider b = p.1 as non zero Element of L by STRUCT_0:def 12;
    take a = p.0, b;
    let n be Element of NAT;
    (n = 0 or ... or n = 1) or n > 1;
    then per cases;
    suppose n = 0 or n = 1;
      hence p.n = <%a,b%>.n by POLYNOM5:38;
    end;
    suppose n > 1;
      then
A2:   1+1 <= n by NAT_1:13;
      hence p.n = 0.L by A1,ALGSEQ_1:8
      .= <%a,b%>.n by A2,POLYNOM5:38;
    end;
  end;
