now :: thesis: for n being Element of NAT st 1 < n & n * n <= 61 & n is prime holds
not n divides 61

let n be Element of NAT ; :: thesis: ( 1 < n & n * n <= 61 & n is prime implies not n divides 61 )
61 = (2 * 30) + 1 ;
then A1: not 2 divides 61 by NAT_4:9;
61 = (3 * 20) + 1 ;
then A2: not 3 divides 61 by NAT_4:9;
61 = (5 * 12) + 1 ;
then A3: not 5 divides 61 by NAT_4:9;
61 = (7 * 8) + 5 ;
then A4: not 7 divides 61 by NAT_4:9;
61 = (11 * 5) + 6 ;
then A5: not 11 divides 61 by NAT_4:9;
61 = (13 * 4) + 9 ;
then A6: not 13 divides 61 by NAT_4:9;
61 = (17 * 3) + 10 ;
then A7: not 17 divides 61 by NAT_4:9;
61 = (19 * 3) + 4 ;
then A8: not 19 divides 61 by NAT_4:9;
61 = (23 * 2) + 15 ;
then A9: not 23 divides 61 by NAT_4:9;
assume ( 1 < n & n * n <= 61 & n is prime ) ; :: thesis: not n divides 61
hence not n divides 61 by A1, A2, A3, A4, A5, A6, A7, A8, A9, NAT_4:62; :: thesis: verum

end;

hence 61 is prime by NAT_4:14; :: thesis: verum