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

theorem Th5:
  for f be Element of the carrier of Polynom-Ring R, n be Nat holds
    ~(f|^n) = (~f)`^n
    proof
      let f be Element of the carrier of Polynom-Ring R, n be Nat;
      set PRR = Polynom-Ring R;
      defpred P[Nat] means ~(f|^$1) = (~f)`^$1;
A2:     f|^0 = 1_PRR by BINOM:8 .= 1.PRR;
        (~f)`^0 = 1_.R by POLYNOM5:15 .= 1.PRR by POLYNOM3:def 10;then
A1:   P[0] by A2;
A3:   for k be Nat holds P[k] implies P[k+1]
      proof
        let k be Nat;
        assume
A4:     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 A4,POLYNOM3:def 10
        .= (~f)`^(k+1) by POLYNOM5:19;
        hence thesis;
      end;
      for k be Nat holds P[k] from NAT_1:sch 2(A1,A3);
      hence thesis;
    end;
