 reserve n for Nat;

theorem
   for R being Ring, S being R-monomorphic R-homomorphic Ring
   for h being Monomorphism of R,S
   for p being Element of the carrier of (Polynom-Ring R)
   holds Roots p = {a where a is Element of R : h.a in Roots (PolyHom h).p}
   proof
     let R be Ring, S be R-monomorphic R-homomorphic Ring;
     let h be Monomorphism of R,S;
     let p be Element of the carrier of (Polynom-Ring R);
A1:   Roots p c={a where a is Element of R : h.a in Roots (PolyHom h).p} by
Th37;
     now let o be object;
       assume o in {a where a is Element of R : h.a in Roots (PolyHom h).p};
       then
       consider a being Element of R such that
A2:    o = a & h.a in Roots (PolyHom h).p;
       h.a is_a_root_of (PolyHom h).p by A2,POLYNOM5:def 10; then
       a is_a_root_of p by Th35;
       hence o in Roots p by A2,POLYNOM5:def 10;
     end;
     hence thesis by TARSKI:2,A1;
   end;
