reserve R for commutative Ring;
reserve A for non degenerated commutative Ring;
reserve I,J,q for Ideal of A;
reserve p for prime Ideal of A;
reserve M,M1,M2 for Ideal of A/q;

theorem Th15:
    for p be prime Ideal of A, n be non zero Nat holds sqrt(p||^n) = p
    proof
      let p be prime Ideal of A, n be non zero Nat;
A1:   p = sqrt p by TOPZARI1:25,TOPZARI1:20;
      defpred P[Nat] means sqrt(p||^$1) = p;
A2:   P[1] by A1,Th8;
A3:   for n be non zero Nat st P[n] holds P[n+1]
      proof
        let n be non zero Nat;
        assume
A4:     P[n];
A5:     n > 0;
        sqrt (p||^(n+1)) = sqrt (p *' (p||^n)) by A5,Th10
        .= sqrt(p /\ (p||^n)) by IDEAL_1:92
        .= sqrt(p) /\ sqrt(p||^n) by Th12
        .= p by A1,A4;
        hence thesis;
      end;
      for i being non zero Nat holds P[i] from NAT_1:sch 10(A2,A3);
      hence thesis;
    end;
