 reserve n for Nat;

theorem Th39:
   for R being Ring, S being R-isomorphic R-homomorphic Ring
   for h being Isomorphism of R,S
   for p being Element of the carrier of (Polynom-Ring R)
   holds Roots (PolyHom h).p = {h.a where a is Element of R : a in Roots p}
   proof
     let R be Ring, S be R-isomorphic R-homomorphic Ring;
     let h be Isomorphism of R,S;
     let p be Element of the carrier of (Polynom-Ring R);
A1:   now let o be object;
       assume
A2:     o in Roots (PolyHom h).p; then
       reconsider b = o as Element of S;
       b is_a_root_of (PolyHom h).p by A2,POLYNOM5:def 10; then
       consider a being Element of R such that
A3:     a is_a_root_of p & h.a = b by Th36;
       a in Roots p by A3,POLYNOM5:def 10;
       hence o in {h.a where a is Element of R : a in Roots p} by A3;
     end;
     now let o be object;
       assume o in {h.a where a is Element of R : a in Roots p}; then
       consider a being Element of R such that
A4:     o = h.a & a in Roots p;
       a is_a_root_of p by A4,POLYNOM5:def 10; then
       h.a is_a_root_of (PolyHom h).p by Th36;
       hence o in Roots (PolyHom h).p by A4,POLYNOM5:def 10;
     end;
     hence thesis by A1,TARSKI:2;
   end;
