reserve k for Nat;
reserve p for Prime;

theorem Ttool331a:
  p < 331 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 or p = 271 or p = 277 or p = 281 or p = 283 or p = 293 or p = 307 or 
  p = 311 or p = 313 or p = 317
  proof
    assume p < 331;
    then 1+1 < p+1 & p < 330+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 317;
      hence thesis by Ttool317a;
    end;
    suppose 317 <= p <= 323+1;
      then 317 <= p <= 317+1 or 318 <= p <= 318+1 or 319 <= p <= 319+1 or 
      320 <= p <= 320+1 or 321 <= p <= 321+1 or 322 <= p <= 322+1 or 
      323 <= p <= 323+1;
      then p = 317 by XPRIMES0:318,319,320,321,322,323,324,NAT_1:9;
      hence thesis;
    end;
    suppose 324 <= p <= 324+1 or 325 <= p <= 325+1 or 
      326 <= p <= 326+1 or 327 <= p <= 327+1 or 328 <= p <= 328+1 or 
      329 <= p <= 329+1;
      then contradiction by XPRIMES0:324,325,326,327,328,329,330,NAT_1:9;
      hence thesis;
    end;
  end;
