 reserve n for Nat;

theorem Th35:
   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), a being Element of R
   holds a is_a_root_of p iff h.a is_a_root_of (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), a be Element of R;
     now assume
A1:    h.a is_a_root_of (PolyHom h).p;
       h.(0.R) = 0.S by RING_2:6
         .= eval((PolyHom h).p,h.a) by A1,POLYNOM5:def 7
         .= h.eval(p,a) by Th28; then
       eval(p,a) = 0.R by FUNCT_2:19;
       hence a is_a_root_of p by POLYNOM5:def 7;
     end;
     hence thesis by Th34;
   end;
