now :: thesis: for n being Element of NAT st 1 < n & n * n <= 89 & n is prime holds
not n divides 89

let n be Element of NAT ; :: thesis: ( 1 < n & n * n <= 89 & n is prime implies not n divides 89 )
89 = (2 * 44) + 1 ;
then A1: not 2 divides 89 by NAT_4:9;
89 = (3 * 29) + 2 ;
then A2: not 3 divides 89 by NAT_4:9;
89 = (5 * 17) + 4 ;
then A3: not 5 divides 89 by NAT_4:9;
89 = (7 * 12) + 5 ;
then A4: not 7 divides 89 by NAT_4:9;
89 = (11 * 8) + 1 ;
then A5: not 11 divides 89 by NAT_4:9;
89 = (13 * 6) + 11 ;
then A6: not 13 divides 89 by NAT_4:9;
89 = (17 * 5) + 4 ;
then A7: not 17 divides 89 by NAT_4:9;
89 = (19 * 4) + 13 ;
then A8: not 19 divides 89 by NAT_4:9;
89 = (23 * 3) + 20 ;
then A9: not 23 divides 89 by NAT_4:9;
assume ( 1 < n & n * n <= 89 & n is prime ) ; :: thesis: not n divides 89
hence not n divides 89 by A1, A2, A3, A4, A5, A6, A7, A8, A9, NAT_4:62; :: thesis: verum

end;

hence 89 is prime by NAT_4:14; :: thesis: verum