 reserve K,F,E for Field,
         R,S for Ring;

theorem
   for S being RingExtension of R,
   p being Element of the carrier of Polynom-Ring R
   holds Roots p c= Roots(S,p)
   proof
     let S be RingExtension of R;
     let p be Element of the carrier of Polynom-Ring R;
A1:  R is Subring of S by Def1; then
A2:  the carrier of R c= the carrier of S by C0SP1:def 3;
     now let o be object;
       assume
A3:    o in Roots p; then
       reconsider a = o as Element of R;
A4:    a is_a_root_of p by A3,POLYNOM5:def 10;
       reconsider b = a as Element of S by A2;
       Ext_eval(p,b) = eval(p,a) by Th14
       .= 0.R by A4,POLYNOM5:def 7
       .= 0.S by A1,C0SP1:def 3; then
       b is_a_root_of p,S;
       hence o in Roots(S,p);
     end;
     hence thesis;
   end;
