reserve k for Nat;
reserve p for Prime;

theorem Ttool29a:
  p < 29 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
  proof
    assume p < 29;
    then 1+1 < p+1 & p < 28+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 23;
      hence thesis by Ttool23a;
    end;
    suppose 23 <= p <= 23+1 or 24 <= p <= 24+1 or 25 <= p <= 25+1 or 
      26 <= p <= 26+1 or 27 <= p <= 27+1;
      then p = 23 by XPRIMES0:24,25,26,27,28,NAT_1:9;
      hence thesis;
    end;
  end;
