 reserve n for Nat;

theorem Th36:
   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),
   b being Element of S holds b is_a_root_of (PolyHom h).p iff
   ex a being Element of R st a is_a_root_of p & h.a = b
   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), b be Element of S;
A1:   now assume A2: b is_a_root_of (PolyHom h).p;
       set a = (h").b;
A3:    dom(h") = the carrier of S by FUNCT_2:def 1;
A4:    h.a = (h"").((h").b) .= b by A3,FUNCT_1:34;
       h.eval(p,a) = eval((PolyHom h).p,h.a) by Th28
              .= 0.S by A4,A2,POLYNOM5:def 7
              .= h.(0.R) by RING_2:6; then
       eval(p,a) = 0.R by FUNCT_2:19;
       hence ex a being Element of R st h.a = b & a is_a_root_of p
         by A4,POLYNOM5:def 7;
     end;
     now assume ex a being Element of R st h.a = b & a is_a_root_of p; then
       consider a being Element of R such that
A5:     h.a = b & a is_a_root_of p;
       eval((PolyHom h).p,b) = h.(eval(p,a)) by A5,Th28
           .= h.(0.R) by A5,POLYNOM5:def 7
           .= 0.S by RING_2:6;
       hence b is_a_root_of (PolyHom h).p by POLYNOM5:def 7;
     end;
     hence thesis by A1;
   end;
