 reserve n for Nat;

theorem Th34:
   for R being Ring, S being R-homomorphic Ring
   for h being Homomorphism of R,S
   for p be Element of the carrier of (Polynom-Ring R), a be Element of R
   holds a is_a_root_of p implies h.a is_a_root_of (PolyHom h).p
   proof
     let R be Ring, S be R-homomorphic Ring; let h be Homomorphism of R,S;
     let p be Element of the carrier of (Polynom-Ring R), a be Element of R;
     assume a is_a_root_of p; then
     eval(p,a) = 0.R by POLYNOM5:def 7; then
     eval((PolyHom h).p, h.a) = h.(0.R) by Th28 .= 0.S by RING_2:6;
     hence thesis by POLYNOM5:def 7;
   end;
