reserve k for Nat;
reserve p for Prime;

theorem Ttool547a:
  p < 547 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 or p = 331 or p = 337 or p = 347 or p = 349 or 
  p = 353 or p = 359 or p = 367 or p = 373 or p = 379 or p = 383 or p = 389 or 
  p = 397 or p = 401 or p = 409 or p = 419 or p = 421 or p = 431 or p = 433 or 
  p = 439 or p = 443 or p = 449 or p = 457 or p = 461 or p = 463 or p = 467 or 
  p = 479 or p = 487 or p = 491 or p = 499 or p = 503 or p = 509 or p = 521 or 
  p = 523 or p = 541
  proof
    assume p < 547;
    then 1+1 < p+1 & p < 546+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 541;
      hence thesis by Ttool541a;
    end;
    suppose 541 <= p <= 541+1 or 542 <= p <= 542+1 or 543 <= p <= 543+1 or 
      544 <= p <= 544+1 or 545 <= p <= 545+1;
      then p = 541 by XPRIMES0:542,543,544,545,546,NAT_1:9;
      hence thesis;
    end;
  end;
