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 Th73:
  for k st k+1 is prime holds k+1 divides a|^(n*k+1) - a
  proof
    let k such that
    A1: k+1 is prime;
    per cases;
    suppose k+1 divides a; then
      k+1 divides a*(a|^(n*k) - 1) by INT_2:2; then
      k+1 divides a*a|^(n*k) - a;
      hence thesis by NEWTON:6;
    end;
    suppose not k+1 divides a; then
      B1: k+1 divides a|^k - 1 by A1,Th59;
      a|^k - 1|^k divides a|^k|^n - 1|^k|^n by NEWTON01:33; then
      k+1 divides  a|^k|^n - 1|^k|^n by B1,INT_2:9; then
      k+1 divides (a|^(n*k) - 1) by NEWTON:9; then
      k+1 divides a*(a|^(n*k) - 1) by INT_2:2; then
      k+1 divides a*a|^(n*k) - a;
      hence thesis by NEWTON:6;
    end;
  end;
