
theorem m4:
for R being Ring,
    S being RingExtension of R
for p being Element of the carrier of Polynom-Ring R
for q being Element of the carrier of Polynom-Ring S
st p = q holds Roots(S,p) = Roots(q)
proof
let R be Ring, S be RingExtension of R;
let p be Element of the carrier of Polynom-Ring R;
let q be Element of the carrier of Polynom-Ring S;
I: Roots(S,p) = {a where a is Element of S : a is_a_root_of p,S}
   by FIELD_4:def 4;
assume AS: p = q;
A: now let o be object;
   assume A0: o in Roots q; then
   reconsider a = o as Element of S;
   a is_a_root_of q by A0,POLYNOM5:def 10; then
   0.S = eval(q,a) by POLYNOM5:def 7
      .= Ext_eval(p,a) by AS,FIELD_4:26; then
   a is_a_root_of p,S by FIELD_4:def 2;
   hence o in Roots(S,p) by I;
   end;
now let o be object;
   assume o in Roots(S,p);
   then consider a being Element of S such that
   A1: o = a & a is_a_root_of p,S by I;
   0.S = Ext_eval(p,a) by A1,FIELD_4:def 2
      .= eval(q,a) by AS,FIELD_4:26; then
   a is_a_root_of q by POLYNOM5:def 7;
   hence o in Roots q by A1,POLYNOM5:def 10;
   end;
hence thesis by A,TARSKI:2;
end;
