reserve k for Nat;
reserve p for Prime;

theorem Ttool257a:
  p < 257 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
  proof
    assume p < 257;
    then 1+1 < p+1 & p < 256+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 251;
      hence thesis by Ttool251a;
    end;
    suppose 251 <= p <= 251+1 or 252 <= p <= 252+1 or 253 <= p <= 253+1 or 
      254 <= p <= 254+1 or 255 <= p <= 255+1;
      then p = 251 by XPRIMES0:252,253,254,255,256,NAT_1:9;
      hence thesis;
    end;
  end;
