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
  2|^m - 1 divides 2|^(2*m+1) - 2 & 2|^m + 1 divides 2|^(2*m+1) - 2
  proof
    2|^(2*m+1) - 2 = 2*2|^(2*m) - 2 by NEWTON:6
    .= (2|^(2*m) - 1|^m)*2
    .= ((2|^m)|^2 - (1|^m)|^2)*2 by NEWTON:9
    .= ((2|^m) - (1|^m))*((2|^m) + (1|^m))*2 by NEWTON01:1
    .= (2|^m - 1)*((2|^m + 1)*2);
    hence thesis by INT_1:def 3,INT_2:2;
  end;
