reserve k for Nat;
reserve p for Prime;

theorem
  p*p <= k < 120409 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
  proof
    assume p*p <= k < 120409;
    then p*p < 347*347 by XXREAL_0:2;
    hence thesis by Ttool347a,NAT_4:1;
  end;
