
theorem lemNor3z:
for F1,F2 being Field,
    p1 being Element of the carrier of Polynom-Ring F1
for p2 being Element of the carrier of Polynom-Ring F2
for E1 being FieldExtension of F1,
    E2 being FieldExtension of F2
st E1 = E2 & p1 = p2 holds Roots(E1,p1) = Roots(E2,p2)
proof
let F1,F2 be Field,
    p1 be Element of the carrier of Polynom-Ring F1;
let p2 be Element of the carrier of Polynom-Ring F2;
let E1 be FieldExtension of F1, E2 be FieldExtension of F2;
assume AS: E1 = E2 & p1 = p2;
the carrier of Polynom-Ring F1 c= the carrier of Polynom-Ring E1 by FIELD_4:10;
then reconsider p1a = p1 as Element of the carrier of Polynom-Ring E1;
the carrier of Polynom-Ring F2 c= the carrier of Polynom-Ring E2 by FIELD_4:10;
then
reconsider p2a = p2 as Element of the carrier of Polynom-Ring E2;
H: Roots(E1,p1) = {a where a is Element of E1 : a is_a_root_of p1,E1} &
   Roots(E2,p2) = {a where a is Element of E2 : a is_a_root_of p2,E2}
   by FIELD_4:def 4;
A: now let o be object;
   assume o in Roots(E1,p1);
   then consider a1 being Element of E1 such that
   B: o = a1 & a1 is_a_root_of p1,E1 by H;
   reconsider a2 = a1 as Element of E2 by AS;
   0.E2 = Ext_eval(p1,a1) by AS,B,FIELD_4:def 2
       .= eval(p1a,a1) by FIELD_4:26
       .= Ext_eval(p2,a2) by AS,FIELD_4:26;
   then a2 is_a_root_of p2,E2 by FIELD_4:def 2;
   hence o in Roots(E2,p2) by B,H;
   end;
now let o be object;
   assume o in Roots(E2,p2);
   then consider a2 being Element of E2 such that
   B: o = a2 & a2 is_a_root_of p2,E2 by H;
   reconsider a1 = a2 as Element of E1 by AS;
   0.E1 = Ext_eval(p2,a2) by AS,B,FIELD_4:def 2
       .= eval(p2a,a2) by FIELD_4:26
       .= Ext_eval(p1,a1) by AS,FIELD_4:26;
   then a1 is_a_root_of p1,E1 by FIELD_4:def 2;
   hence o in Roots(E1,p1) by B,H;
   end;
hence thesis by A,TARSKI:2;
end;
