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 p be prime Nat st not p divides a holds
    ex n st p divides a|^n - 1 & 0 < n & n < p
  proof
    let p be prime Nat such that
    A0: not p divides a;
    p > 1 by INT_2:def 4; then
    p >= 1+1 by NAT_1:13; then
    consider k such that
    A1: p = 2 + k by NAT_1:10;
    p = (k+1) + 1 by A1; then
    p divides (a|^(k+1) - 1) by A0,Th59;
    hence thesis by A1,XREAL_1:6;
  end;
