 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;
 reserve m for maximal Ideal of A;
 reserve p for prime Ideal of A;
 reserve k for non zero Nat;

theorem Th23:
  not a in p implies not a|^k in p
  proof
    assume
A1: not a in p;
    not a|^k in p
    proof
      defpred P[Nat] means not a|^$1 in p;
A2:   P[1] by A1,BINOM:8;
A3:   for k holds P[k] implies P[k+1]
      proof
        let k;
        assume
A4:     P[k];
A5:     a|^(k+1) = (a|^k)*(a|^1) by BINOM:10;
        not (a|^1) in p by A1,BINOM:8;
        hence thesis by A4,A5,RING_1:def 1;
      end;
      for k holds P[k] from NAT_1:sch 10(A2,A3);
      hence thesis;
    end;
    hence thesis;
  end;
