now :: thesis: for n being Element of NAT st 1 < n & n * n <= 31 & n is prime holds
not n divides 31

let n be Element of NAT ; :: thesis: ( 1 < n & n * n <= 31 & n is prime implies not n divides 31 )
31 = (2 * 15) + 1 ;
then A1: not 2 divides 31 by NAT_4:9;
31 = (3 * 10) + 1 ;
then A2: not 3 divides 31 by NAT_4:9;
31 = (5 * 6) + 1 ;
then A3: not 5 divides 31 by NAT_4:9;
31 = (7 * 4) + 3 ;
then A4: not 7 divides 31 by NAT_4:9;
31 = (11 * 2) + 9 ;
then A5: not 11 divides 31 by NAT_4:9;
31 = (13 * 2) + 5 ;
then A6: not 13 divides 31 by NAT_4:9;
31 = (17 * 1) + 14 ;
then A7: not 17 divides 31 by NAT_4:9;
31 = (19 * 1) + 12 ;
then A8: not 19 divides 31 by NAT_4:9;
31 = (23 * 1) + 8 ;
then A9: not 23 divides 31 by NAT_4:9;
assume ( 1 < n & n * n <= 31 & n is prime ) ; :: thesis: not n divides 31
hence not n divides 31 by A1, A2, A9, A8, A4, A3, A6, A5, A7, NAT_4:62; :: thesis: verum

end;

hence 31 is prime by NAT_4:14; :: thesis: verum