 reserve n for Nat;

theorem Th3:
  for R being Ring,
      I being Ideal of R
  for x being Element of R/I,
      a being Element of R st x = Class(EqRel(R,I),a)
  for n being Nat holds x|^n = Class(EqRel(R,I),a|^n)
  proof
    let R be Ring, I be Ideal of R;
    let x be Element of R/I, a be Element of R;
    assume
A1: x = Class(EqRel(R,I),a);
    let n be Nat;
    defpred P[Nat] means x|^($1) = Class(EqRel(R,I),a|^($1));
    x|^0 = 1_(R/I) by BINOM:8 .= Class(EqRel(R,I),1_R) by RING_1:def 6
        .= Class(EqRel(R,I),a|^0) by BINOM:8; then
A2: P[0];
A3: now let i be Nat;
    assume
A4: P[i];
    Class(EqRel(R,I),a|^(i+1)) = Class(EqRel(R,I),(a|^i)*(a|^1)) by BINOM:10
      .= Class(EqRel(R,I),(a|^i)*a) by BINOM:8
      .= (x|^i) * x by A1,A4,RING_1:14
      .= (x|^i) * (x|^1) by BINOM:8
      .= x|^(i+1) by BINOM:10;
    hence P[i+1];
    end;
    for i being Nat holds P[i] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
