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;

theorem
  6 divides a|^3 - a
  proof
    3 divides a*1 or 3 divides (a+1)*(a-1) by Th50; then
    A1: 3 divides (a-1)*(a+1)*(a*1) by INT_2:2;
    a is even or (a+1) is even; then
    A2: a*(a+1)*(a-1) is even;
    2*1+1 is odd; then
    A3: 3,2|^1 are_coprime by NAT_5:3;
    a|^(2+1) - a = a|^2*a - a by NEWTON:6
    .= (a|^2 -1|^2)*a*1
    .= (a-1)*(a+1)*a*1 by NEWTON01:1; then
    |.3.| divides |.a|^3 - a.| & |.2.| divides |.a|^3 - a.|
      by A1,A2,INT_2:16; then
    |.3*2.| divides |.a|^3 - a.| by PEPIN:4,A3;
    hence thesis by INT_2:16;
  end;
