 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;

theorem Th4:
  for f be Element of the carrier of Polynom-Ring INT.Ring, n be Nat holds
    ^(f|^n) = (^ f)|^n
   proof
     let f be Element of the carrier of Polynom-Ring INT.Ring, n be Nat;
     set PRI = Polynom-Ring INT.Ring;
     set PRF = Polynom-Ring F_Real;
     defpred P[Nat] means ^(f|^$1) = (^ f)|^$1;
A1:  P[0]
     proof
A2:     PRI is Subring of PRF by FIELD_4:def 1;
A3:     f|^0 = 1_PRI by BINOM:8 .= 1.PRI;
        (^ f)|^0 = 1_PRF by BINOM:8 .= 1.PRF;
        hence thesis by A3,A2,C0SP1:def 3;
     end;
A4:  for k be Nat holds P[k] implies P[k+1]
     proof
       let k be Nat;
       assume
A5:    P[k];
       f|^(k+1) = (f|^k)*(f|^1) by BINOM:10 .= (f|^k)*f by BINOM:8; then
       ^(f|^(k+1)) = ((^ f)|^k)*(^ f) by A5,E_TRANS1:27
         .= ((^ f)|^k)*((^ f)|^1) by BINOM:8 .= (^ f)|^(k+1) by BINOM:10;
       hence thesis;
     end;
     for k be Nat holds P[k] from NAT_1:sch 2(A1,A4);
     hence thesis;
   end;
