reserve n for Nat;

theorem
for F being domRing,
    p being Polynomial of F
for b being non zero Element of F holds Roots(b * p) = Roots p
proof
let R be domRing, p be Polynomial of R; let b be non zero Element of R;
A: now let o be object;
   assume B0: o in Roots p;
   then reconsider a = o as Element of R;
   a is_a_root_of p by B0,POLYNOM5:def 10;
   then 0.R = eval(p,a) by POLYNOM5:def 7;
   then eval(b*p,a) = b * 0.R by Th30 .= 0.R;
   then a is_a_root_of (b*p) by POLYNOM5:def 7;
   hence o in Roots(b*p) by POLYNOM5:def 10;
   end;
now let o be object;
   assume B0: o in Roots(b*p);
   then reconsider a = o as Element of R;
   a is_a_root_of (b*p) by B0,POLYNOM5:def 10;
   then 0.R = eval(b*p,a) by POLYNOM5:def 7
           .= b * eval(p,a) by Th30;
   then eval(p,a) = 0.R by VECTSP_2:def 1;
   then a is_a_root_of p by POLYNOM5:def 7;
   hence o in Roots p by POLYNOM5:def 10;
  end;
hence thesis by A,TARSKI:2;
end;
