reserve n for Nat;

theorem div100:
for R being domRing,
    a,b being Element of R holds rpoly(1,a) divides rpoly(1,b) iff a = b
proof
let R be domRing, a,b be Element of R;
X: now assume rpoly(1,a) divides rpoly(1,b);
   then consider p being Polynomial of R such that
   A: rpoly(1,a) *' p = rpoly(1,b) by RING_4:1;
   B: {b} = Roots rpoly(1,b) by ro4 .= Roots rpoly(1,a) \/ Roots p
   by A,UPROOTS:23;
   a in {a} by TARSKI:def 1;
   then a in Roots rpoly(1,a) by ro4;
   then a in {b} by B,XBOOLE_0:def 3;
   hence a = b by TARSKI:def 1;
   end;
now assume a = b;
  then rpoly(1,a) *' 1_.(R) = rpoly(1,b);
  hence rpoly(1,a) divides rpoly(1,b) by RING_4:1;
  end;
hence thesis by X;
end;
