 reserve n for Nat;

theorem Th23:
   for R being Ring, S being R-homomorphic Ring
   for h being Homomorphism of R,S holds (PolyHom h).(0_.R) = 0_.S
   proof
     let R be Ring, S be R-homomorphic Ring;
     let h be Homomorphism of R,S;
A1:   0_.R = 0.(Polynom-Ring R) by POLYNOM3:def 10;
      reconsider f = (PolyHom h).(0.(Polynom-Ring R)) as
         Element of the carrier of Polynom-Ring S;
     now let i be Element of NAT;
       f.i = h.((0_.R).i) by Def2,A1
       .= h.(0.R) by FUNCOP_1:7
       .= 0.S by RING_2:6
       .= (0_.S).i by FUNCOP_1:7;
       hence f.i = (0_.S).i;
     end;
     hence thesis by A1,FUNCT_2:63;
   end;
