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 Th59:
  for k be Nat st k+1 is prime & not k+1 divides a
    holds k+1 divides a|^k - 1
  proof
    let k be Nat such that
    A1: k+1 is prime and
    A2: not k+1 divides a;
    k+1 divides a|^(k+1) - a by A1,Th58; then
    k+1 divides a|^k*a - a by NEWTON:6; then
    k+1 divides a*(a|^k - 1);
    hence thesis by A1,A2,INT_5:7;
  end;
