reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th16:
  for p being Prime, n being Nat
  holds card { k where k is Nat: k divides p|^n } = n + 1
  proof
    let p be Prime;
    defpred P[Nat] means card { k where k is Nat: k divides p|^$1 } = $1 + 1;
  set X = { k where k is Nat: k divides p|^0 };
A1: p|^0 = 1 by NEWTON:4;
  1 divides p|^0 by NAT_D:6;
  then
A2: 1 in X;
  X c= {1}
  proof
    let x be object;
    assume x in X;
    then consider k be Nat such that
A3: k=x & k divides p|^0;
    k = 1 or k=-1 by A1,A3,INT_2:13;
    hence thesis by A3,TARSKI:def 1;
  end;
  then X={1} by A2,ZFMISC_1:33;
  then
A4: P[0] by CARD_1:30;
A5: for i being Nat holds P[i] implies P[i+1]
    proof
      let i be Nat;
      assume
A6:   P[i];
      set i1=i+1;
      set X1 = { k where k is Nat: k divides p|^i1 };
      reconsider X = { k where k is Nat: k divides p|^i} as finite set by A6;
A7:   X\/{p|^i1} = X1 by Lm5;
      then not p|^i1 in X by Lm5;
      hence thesis by A7,A6,CARD_2:41;
    end;
    for i being Nat holds P[i] from NAT_1:sch 2(A4,A5);
    hence thesis;
  end;
