
theorem Th8:
  for x,p being Prime,
      k being non zero Nat holds x divides (p|^k) iff x = p
proof
  let x,p be Prime;
  let k be non zero Nat;
A1: now assume A2: x divides (p|^k);
   defpred P[Nat] means x divides p|^($1) implies x = p;
   A3: P[1]
      proof
      assume x divides p|^1;
      then x divides p;
      then x = 1 or x = p by INT_2:def 4;
      hence x = p by NAT_2:def 1;
      end;
   A4: now let k be non zero Nat;
      assume A5: P[k];
      now assume A6: x divides p|^(k+1);
     A7: p|^(k+1) = p * p|^k by NEWTON:6;
        per cases by INT_2:30;
        suppose x,p are_coprime;
          hence x = p by A5,A6,A7,INT_2:25;
          end;
        suppose x = p;
          hence x = p;
          end;
        end;
      hence P[k + 1];
      end;
   A8: for k being non zero Nat holds P[k] from NAT_1:sch 10(A3,A4);
   thus x = p by A8,A2;
   end;
now assume A9: x = p;
   reconsider k1 = k-1 as Nat;
   p * p|^k1 = p|^(k1+1) by NEWTON:6;
   hence x divides (p|^k) by A9;
   end;
hence thesis by A1;
end;
