 reserve n for Nat;

theorem Th37:
   for R being Ring, S being R-homomorphic Ring
   for h being Homomorphism of R,S
   for p being Element of the carrier of (Polynom-Ring R) holds
   Roots p c= {a where a is Element of R : h.a in Roots (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);
     let o be object;
     assume
A1:  o in Roots p; then
     reconsider a = o as Element of R;
     a is_a_root_of p by A1,POLYNOM5:def 10; then
     h.a is_a_root_of (PolyHom h).p by Th34; then
     h.a in Roots (PolyHom h).p by POLYNOM5:def 10;
     hence o in {a where a is Element of R : h.a in Roots (PolyHom h).p};
   end;
