reserve k for Nat;
reserve p for Prime;

theorem Ttool31a:
  p < 31 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
  proof
    assume p < 31;
    then 1+1 < p+1 & p < 30+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 29;
      hence thesis by Ttool29a;
    end;
    suppose 29 <= p <= 29+1;
      hence thesis by XPRIMES0:30,NAT_1:9;
    end;
  end;
