
theorem qua5:
for R being non degenerated Ring,
    p being Polynomial of R holds
p is quadratic iff
ex a being non zero Element of R, b,c being Element of R st p = <%c,b,a%>
proof
let R be non degenerated Ring, p be Polynomial of R;
now assume A: p is quadratic;
  B: deg p = len p - 1 by HURWITZ:def 2;
  reconsider a = p.2, b = p.1, c = p.0 as Element of R;
  now let i be Element of NAT;
    i <= 2 implies i = 0 or ... or i = 2; then
    per cases;
    suppose i = 0;
      hence p.i = <%c,b,a%>.i by qua1;
      end;
    suppose i = 1;
      hence p.i = <%c,b,a%>.i by qua1;
      end;
    suppose i = 2;
      hence p.i = <%c,b,a%>.i by qua1;
      end;
    suppose i > 2;
      then i + 1 > 2 + 1 by XREAL_1:6;
      then C: i >= 3 by NAT_1:13;
      hence <%c,b,a%>.i = 0.R by qua1 .= p.i by A,B,C,ALGSEQ_1:8;
      end;
    end; then
  D: p = <%c,b,a%>;
  len p = 2 + 1 by B,A;
  then a is non zero by ALGSEQ_1:10;
  hence ex a being non zero Element of R,
                             b,c being Element of R st p = <%c,b,a%> by D;
  end;
hence thesis;
end;
