reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem
  for n be prime Nat holds n divides a|^(n+k) - a|^(k+1)
  proof
    let n be prime Nat;
    a|^(n+k) - a|^(k+1) = a|^n *a|^k - a|^(k+1) by NEWTON:8
    .= a|^n * a|^k - a|^k *a by NEWTON:6
    .= a|^k*(a|^n - a);
    hence thesis by Th58,INT_2:2;
  end;
