
theorem prm:
for R being Ring,
    S being RingExtension of R
for p being Element of the carrier of Polynom-Ring R
for h being R-fixing Monomorphism of S
for a being Element of S holds a in Roots(S,p) iff h.a in Roots(S,p)
proof
let F be Ring, E be RingExtension of F;
let p be Element of the carrier of Polynom-Ring F;
let h be F-fixing Monomorphism of E;
let a be Element of E;
h is monomorphism; then
reconsider E1 = E as E-monomorphic E-homomorphic Ring
  by RING_2:def 4,RING_3:def 3;
reconsider h1 = h as Monomorphism of E,E1;
the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E by FIELD_4:10;
then reconsider p1 = p as Element of the carrier of Polynom-Ring E;
now let i be Element of NAT;
  thus p.i = h1.(p.i) by deffix .= ((PolyHom h1).p1).i by FIELD_1:def 2;
  end;
then H: (PolyHom h1).p1 = p1;
A: now assume a in Roots(E,p);
   then a in Roots p1 by FIELD_7:13;
   then a is_a_root_of p1 by POLYNOM5:def 10;
   then h1.a is_a_root_of (PolyHom h1).p1 by FIELD_1:34;
   then h.a in Roots p1 by H,POLYNOM5:def 10;
   hence h.a in Roots(E,p) by FIELD_7:13;
   end;
now assume h.a in Roots(E,p); 
   then A0: h.a in Roots (PolyHom h1).p1 by H,FIELD_7:13;
   Roots p1 = {a where a is Element of E : h1.a in Roots (PolyHom h1).p1} 
      by FIELD_1:37;
   then a in Roots p1 by A0;
   hence a in Roots(E,p) by FIELD_7:13;
   end;
hence thesis by A;
end;
