reserve k for Nat;
reserve p for Prime;

theorem Ttool61a:
  p < 61 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 or p = 59
  proof
    assume p < 61;
    then 1+1 < p+1 & p < 60+1 by XREAL_1:6,INT_2:def 4;
    then per cases by NAT_1:13;
    suppose 2 <= p < 59;
      hence thesis by Ttool59a;
    end;
    suppose 59 <= p <= 59+1;
      then p = 59 by XPRIMES0:60,NAT_1:9;
      hence thesis;
    end;
  end;
