
theorem deg2:
for R being Ring,
    p being Polynomial of R st deg p < 2
for a being Element of R ex y,z being Element of R st p = <%y,z%>
proof
let R be Ring, p be Polynomial of R;
assume A: deg p < 2;
let a be Element of R;
take y = p.0, z = p.1;
set q = <%y,z%>;
now let i be Element of NAT;
  per cases;
  suppose i = 0;
    hence p.i = q.i by POLYNOM5:38;
    end;
  suppose i = 1;
    hence p.i = q.i by POLYNOM5:38;
    end;
  suppose i <> 0 & i <> 1; then
    B: i is non trivial by NAT_2:def 1; then
    i >= 2 by NAT_2:29; then
    i > deg p by A,XXREAL_0:2; then
    i > len p - 1 by HURWITZ:def 2; then
    i >= len p - 1 + 1 by INT_1:7;
    hence p.i = 0.R by ALGSEQ_1:8 .= q.i by B,NAT_2:29,POLYNOM5:38;
    end;
  end;
hence thesis;
end;
