1459 = (2 * 729) + 1 ;
hence not 2 divides 1459 by NAT_4:9; :: thesis:
1459 = (3 * 486) + 1 ;
hence not 3 divides 1459 by NAT_4:9; :: thesis:
1459 = (5 * 291) + 4 ;
hence not 5 divides 1459 by NAT_4:9; :: thesis:
1459 = (7 * 208) + 3 ;
hence not 7 divides 1459 by NAT_4:9; :: thesis:
1459 = (11 * 132) + 7 ;
hence not 11 divides 1459 by NAT_4:9; :: thesis:
1459 = (13 * 112) + 3 ;
hence not 13 divides 1459 by NAT_4:9; :: thesis:
1459 = (17 * 85) + 14 ;
hence not 17 divides 1459 by NAT_4:9; :: thesis:
1459 = (19 * 76) + 15 ;
hence not 19 divides 1459 by NAT_4:9; :: thesis:
1459 = (23 * 63) + 10 ;
hence not 23 divides 1459 by NAT_4:9; :: thesis:
1459 = (29 * 50) + 9 ;
hence not 29 divides 1459 by NAT_4:9; :: thesis:
1459 = (31 * 47) + 2 ;
hence not 31 divides 1459 by NAT_4:9; :: thesis:
1459 = (37 * 39) + 16 ;
hence not 37 divides 1459 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1459 & n is prime holds
not n divides 1459 by XPRIMET1:24;
hence 1459 is prime by NAT_4:14; :: thesis: