
theorem
  for n being non zero Nat ex x being non zero Nat st
  for i being Nat holds n divides x|^|^(i+1) + 1
  proof
    let n be non zero Nat;
    A1: n divides 2*n & 2 divides 2*n;
    set x=2*n-1;
    reconsider x as non zero odd Nat;
    take x;
    let i be Nat;

    A3: x|^|^((i)+1) = x|^|^(Segm((i)+1))
    .= x|^|^(succ Segm(i)) by NAT_1:38
    .= exp(x,x|^|^(i)) by ORDINAL5:14
    .= x|^(x|^|^(i));

    2*n divides x - -1; then
    x|^(x|^|^(i)), (-1)|^(x|^|^(i)) are_congruent_mod 2*n
    by GR_CY_3:34,INT_1:def 4;
    then x|^|^(i+1),-1 are_congruent_mod 2*n by A3;
    hence thesis by A1,INT_2:9;
  end;
