reserve n for Nat;

theorem repr:
for R being non degenerated well-unital non empty doubleLoopStr
for a being Element of R holds <%-a, 1.R%> = rpoly(1,a)
proof
let R be non degenerated well-unital non empty doubleLoopStr,
    a be Element of R;
set p = <%-a, 1.R%>, q = rpoly(1,a);
A: 1 = deg q by HURWITZ:27 .= len q - 1 by HURWITZ:def 2;
D: 1.R <> 0.R;
now let k be Nat;
  assume k < len p;
  then k < 1 + 1 by D,POLYNOM5:40;
  then B: k <= 1 by NAT_1:13;
  per cases by B,NAT_1:25;
  suppose C: k = 0;
    hence p.k = -(1_R * a) by POLYNOM5:38
             .= -(power(R).(a,0) * a) by GROUP_1:def 7
             .= -power(R).(a,0+1) by GROUP_1:def 7
             .= q.k by C,HURWITZ:25;
    end;
  suppose C: k = 1;
    hence p.k = 1_R by POLYNOM5:38 .= q.k by C,HURWITZ:25;
    end;
  end;
hence thesis by A,D,POLYNOM5:40,ALGSEQ_1:12;
end;
