
theorem
  for L being non trivial ZeroStr, p being Polynomial of L st len p = 1
  ex a being non zero Element of L st p = <%a%>
  proof
    let L be non trivial ZeroStr;
    let p be Polynomial of L;
    assume
A1: len p = 1;
    1 = 1+0;
    then p.0 <> 0.L by A1,ALGSEQ_1:10;
    then reconsider a = p.0 as non zero Element of L by STRUCT_0:def 12;
    take a;
    let n be Element of NAT;
    per cases;
    suppose n = 0;
      hence p.n = <%a%>.n by POLYNOM5:32;
    end;
    suppose n > 0;
      then
A2:   0+1 <= n by NAT_1:13;
      hence p.n = 0.L by A1,ALGSEQ_1:8
      .= <%a%>.n by A2,POLYNOM5:32;
    end;
  end;
