reserve k for Nat;
reserve p for Prime;

theorem Ttool37a:
  p < 37 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
  proof
    assume p < 37;
    then 1+1 < p+1 & p < 36+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 31;
      hence thesis by Ttool31a;
    end;
    suppose 31 <= p <= 31+1 or 32 <= p <= 32+1 or 33 <= p <= 33+1 or 
      34 <= p <= 34+1 or 35 <= p <= 35+1;
      then p = 31 by XPRIMES0:32,33,34,35,36,NAT_1:9;
      hence thesis;
    end;
  end;
