theorem Th37:
    for x be Element of Polynom-Ring R, f be Polynomial of R
    st x = f holds for n be Nat holds x|^n = f`^n
    proof
      let x be Element of Polynom-Ring R, f be Polynomial of R;
      assume
A1:   x = f;
      defpred P[Nat] means x|^$1 = f`^$1;
A2:   for n be Nat st P[n] holds P[n+1]
      proof
        let n be Nat;
        assume
A3:     P[n];
        x|^(n+1) = (x|^n) * (x|^1) by BINOM:10 .= (x|^n) * x by BINOM:8
        .= (f`^n) *' f by A1,A3,POLYNOM3:def 10
        .= f`^(n+1) by POLYNOM5:19;
        hence thesis;
      end;
      x|^0 = 1_(Polynom-Ring R) by BINOM:8 .= 1_.(R) by POLYNOM3:37
      .= f`^0 by POLYNOM5:15; then
A4:   P[0];
      for n be Nat holds P[n] from NAT_1:sch 2(A4,A2);
      hence thesis;
    end;
