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

theorem
  n > 0 implies 2|^(2|^(n+1)) + 2|^(2|^n) + 1 >= 21
  proof
    assume n > 0;
    then
A1: n >= 0+1 by NAT_1:13;
    then n+1 >= 1+1 by XREAL_1:6;
    then 2|^(n+1) >= 2|^2 by PREPOWER:93;
    then
A2: 2|^(2|^(n+1)) >= 2|^4 by Lm2,PREPOWER:93;
    2|^n >= 2|^1 by A1,PREPOWER:93;
    then 2|^(2|^n) >= 2|^2 by PREPOWER:93;
    then 2|^(2|^(n+1)) + 2|^(2|^n) >= 16 + 4 by A2,Lm2,Lm4,XREAL_1:7;
    then 2|^(2|^(n+1)) + 2|^(2|^n) + 1 >= 20 + 1 by XREAL_1:6;
    hence thesis;
  end;
