reserve a,b,c,k,m,n for Nat;
reserve p for Prime;

theorem
  2|^(2|^n)+1 divides 2|^(2|^(2|^n)+1)-2
  proof
    set A = 2|^(2|^n);
    set F = A+1;
    set G = 2|^(2|^(n+1))-1;
    set H = 2|^A-1;
    set I = 2|^(A+1);
    n+1 <= 2|^n by NEWTON:85;
    then
A1: G divides H by Th19,NEWTON:89;
    (A+1)*(A-1) = A|^2-1|^2 by NEWTON01:1
    .= 2|^(2|^n*2|^1)-1 by NEWTON:9
    .= 2|^(2|^(n+1))-1 by NEWTON:8;
    then F divides G;
    then
A2: F divides H by A1,INT_2:9;
    2|^1*2|^A = I by NEWTON:8;
    then 2*H = I-2;
    then H divides I-2;
    hence F divides 2|^F-2 by A2,INT_2:9;
  end;
