reserve k for Nat;
reserve p for Prime;

theorem Ttool23a:
  p < 23 implies
  p = 2 or p = 3 or p = 5 or p = 7 or p = 11 or p = 13 or p = 17 or 
  p = 19
  proof
    assume p < 23;
    then 1+1 < p+1 & p < 22+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 19;
      hence thesis by Ttool19a;
    end;
    suppose 19 <= p <= 19+1 or 20 <= p <= 20+1 or 21 <= p <= 21+1;
      then p = 19 by XPRIMES0:20,21,22,NAT_1:9;
      hence thesis;
    end;
  end;
