
theorem qua5a:
for R being non degenerated Ring,
    p being monic Polynomial of R holds
p is quadratic iff ex b,c being Element of R st p = <%c,b,1.R%>
proof
let R be non degenerated Ring, p be monic Polynomial of R;
now assume p is quadratic; then
  consider a being non zero Element of R, b,c being Element of R such that
  A: p = <%c,b,a%> by qua5;
  H: 3-'1 = 3-1 & a <> 0.R by XREAL_0:def 2;
  1.R = LC p by RATFUNC1:def 7
     .= p.(len p-'1) by RATFUNC1:def 6
     .= p.2 by H,A,qua3
     .= a by A,qua1;
  hence ex b,c being Element of R st p = <%c,b,1.R%> by A;
  end;
hence thesis;
end;
