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

theorem Th39:
  2|^(2|^(n+1)) + 2|^(2|^n) + 1 >= 7
  proof
    n+1 >= 0+1 by XREAL_1:6;
    then 2|^(n+1) >= 2|^1 by PREPOWER:93;
    then
A1: 2|^(2|^(n+1)) >= 2|^2 by PREPOWER:93;
    2|^n >= 2|^0 by PREPOWER:93;
    then 2|^(2|^n) >= 2|^(2|^0) by PREPOWER:93;
    then 2|^(2|^n) >= 2|^1 by NEWTON:4;
    then 2|^(2|^(n+1)) + 2|^(2|^n) >= 4 + 2 by A1,Lm2,XREAL_1:7;
    then 2|^(2|^(n+1)) + 2|^(2|^n) + 1 >= 6 + 1 by XREAL_1:6;
    hence thesis;
  end;
