reserve k for Nat;
reserve p for Prime;

theorem Ttool223a:
  p < 223 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
  proof
    assume p < 223;
    then 1+1 < p+1 & p < 222+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 211;
      hence thesis by Ttool211a;
    end;
    suppose 211 <= p <= 211+1 or 212 <= p <= 212+1 or 213 <= p <= 213+1 or 
      214 <= p <= 214+1 or 215 <= p <= 215+1 or 216 <= p <= 216+1 or 
      217 <= p <= 217+1 or 218 <= p <= 218+1 or 219 <= p <= 219+1 or 
      220 <= p <= 220+1 or 221 <= p <= 221+1;
      then p = 211 by XPRIMES0:212,213,214,215,216,217,218,219,220,221,222,
        NAT_1:9;
      hence thesis;
    end;
  end;
