2089 = (2 * 1044) + 1 ;
hence not 2 divides 2089 by NAT_4:9; :: thesis:
2089 = (3 * 696) + 1 ;
hence not 3 divides 2089 by NAT_4:9; :: thesis:
2089 = (5 * 417) + 4 ;
hence not 5 divides 2089 by NAT_4:9; :: thesis:
2089 = (7 * 298) + 3 ;
hence not 7 divides 2089 by NAT_4:9; :: thesis:
2089 = (11 * 189) + 10 ;
hence not 11 divides 2089 by NAT_4:9; :: thesis:
2089 = (13 * 160) + 9 ;
hence not 13 divides 2089 by NAT_4:9; :: thesis:
2089 = (17 * 122) + 15 ;
hence not 17 divides 2089 by NAT_4:9; :: thesis:
2089 = (19 * 109) + 18 ;
hence not 19 divides 2089 by NAT_4:9; :: thesis:
2089 = (23 * 90) + 19 ;
hence not 23 divides 2089 by NAT_4:9; :: thesis:
2089 = (29 * 72) + 1 ;
hence not 29 divides 2089 by NAT_4:9; :: thesis:
2089 = (31 * 67) + 12 ;
hence not 31 divides 2089 by NAT_4:9; :: thesis:
2089 = (37 * 56) + 17 ;
hence not 37 divides 2089 by NAT_4:9; :: thesis:
2089 = (41 * 50) + 39 ;
hence not 41 divides 2089 by NAT_4:9; :: thesis:
2089 = (43 * 48) + 25 ;
hence not 43 divides 2089 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2089 & n is prime holds
not n divides 2089 by XPRIMET1:28;
hence 2089 is prime by NAT_4:14; :: thesis: