 reserve n for Nat;

theorem Th14:
   for R being Ring, S being R-homomorphic Ring
   for h being Homomorphism of R,S
   for a being Element of R, n being Nat holds h.(a|^n) = (h.a)|^n
   proof
     let R be Ring, S be R-homomorphic Ring; let h be Homomorphism of R,S;
     let a be Element of R, n be Nat;
     defpred P[Nat] means h.(a|^($1)) = (h.a)|^($1);
A1:  P[0]
     proof
       thus
       h.(a|^0) = h.(1_R) by BINOM:8 .= 1_S by GROUP_1:def 13
       .= (h.a)|^0 by BINOM:8;
     end;
A2:  now let k be Nat;
       assume
A3:    P[k];
       h.(a|^(k+1)) = h.((a|^k) * (a|^1)) by BINOM:10
                .= h.(a|^k) * h.(a|^1) by GROUP_6:def 6
                .= (h.a)|^k * (h.a) by A3,BINOM:8
                .= (h.a)|^k * ((h.a)|^1) by BINOM:8
                .= (h.a)|^(k+1) by BINOM:10;
       hence P[k+1];
     end;
     for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
     hence thesis;
    end;
