reserve k for Nat;
reserve p for Prime;

theorem Ttool59a:
  p < 59 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
  proof
    assume p < 59;
    then 1+1 < p+1 & p < 58+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 53;
      hence thesis by Ttool53a;
    end;
    suppose 53 <= p <= 53+1 or 54 <= p <= 54+1 or 55 <= p <= 55+1 or 
      56 <= p <= 56+1 or 57 <= p <= 57+1;
      then p = 53 by XPRIMES0:54,55,56,57,58,NAT_1:9;
      hence thesis;
    end;
  end;
