reserve k for Nat;
reserve p for Prime;

theorem Ttool271a:
  p < 271 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or 
  p = 19 or p = 23 or p = 29 or p = 31 or p = 37 or p = 41 or p = 43 or 
  p = 47 or p = 53 or p = 59 or p = 61 or p = 67 or p = 71 or p = 73 or 
  p = 79 or p = 83 or p = 89 or p = 97 or p = 101 or p = 103 or p = 107 or 
  p = 109 or p = 113 or p = 127 or p = 131 or p = 137 or p = 139 or p = 149 or 
  p = 151 or p = 157 or p = 163 or p = 167 or p = 173 or p = 179 or p = 181 or 
  p = 191 or p = 193 or p = 197 or p = 199 or p = 211 or p = 223 or p = 227 or 
  p = 229 or p = 233 or p = 239 or p = 241 or p = 251 or p = 257 or p = 263 or 
  p = 269
  proof
    assume p < 271;
    then 1+1 < p+1 & p < 270+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 269;
      hence thesis by Ttool269a;
    end;
    suppose 269 <= p <= 269+1;
      then p = 269 by XPRIMES0:270,NAT_1:9;
      hence thesis;
    end;
  end;
