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

theorem Th41:
  n > 1 implies 2|^(2|^(n+1)) + 2|^(2|^n) + 1 >= 273
  proof
    assume
A1: n > 1;
    then n+1 > 1+1 by XREAL_1:6;
    then n+1 >= 2+1 by NAT_1:13;
    then 2|^(n+1) >= 2|^3 by PREPOWER:93;
    then
A2: 2|^(2|^(n+1)) >= 2|^8 by Lm3,PREPOWER:93;
    n >= 1+1 by A1,NAT_1:13;
    then 2|^n >= 2|^2 by PREPOWER:93;
    then 2|^(2|^n) >= 2|^4 by Lm2,PREPOWER:93;
    then 2|^(2|^(n+1)) + 2|^(2|^n) >= 256 + 16 by A2,Lm4,Lm6,XREAL_1:7;
    then 2|^(2|^(n+1)) + 2|^(2|^n) + 1 >= 272 + 1 by XREAL_1:6;
    hence thesis;
  end;
