3089 = (2 * 1544) + 1 ;
hence not 2 divides 3089 by NAT_4:9; :: thesis:
3089 = (3 * 1029) + 2 ;
hence not 3 divides 3089 by NAT_4:9; :: thesis:
3089 = (5 * 617) + 4 ;
hence not 5 divides 3089 by NAT_4:9; :: thesis:
3089 = (7 * 441) + 2 ;
hence not 7 divides 3089 by NAT_4:9; :: thesis:
3089 = (11 * 280) + 9 ;
hence not 11 divides 3089 by NAT_4:9; :: thesis:
3089 = (13 * 237) + 8 ;
hence not 13 divides 3089 by NAT_4:9; :: thesis:
3089 = (17 * 181) + 12 ;
hence not 17 divides 3089 by NAT_4:9; :: thesis:
3089 = (19 * 162) + 11 ;
hence not 19 divides 3089 by NAT_4:9; :: thesis:
3089 = (23 * 134) + 7 ;
hence not 23 divides 3089 by NAT_4:9; :: thesis:
3089 = (29 * 106) + 15 ;
hence not 29 divides 3089 by NAT_4:9; :: thesis:
3089 = (31 * 99) + 20 ;
hence not 31 divides 3089 by NAT_4:9; :: thesis:
3089 = (37 * 83) + 18 ;
hence not 37 divides 3089 by NAT_4:9; :: thesis:
3089 = (41 * 75) + 14 ;
hence not 41 divides 3089 by NAT_4:9; :: thesis:
3089 = (43 * 71) + 36 ;
hence not 43 divides 3089 by NAT_4:9; :: thesis:
3089 = (47 * 65) + 34 ;
hence not 47 divides 3089 by NAT_4:9; :: thesis:
3089 = (53 * 58) + 15 ;
hence not 53 divides 3089 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3089 & n is prime holds
not n divides 3089 by XPRIMET1:32;
hence 3089 is prime by NAT_4:14; :: thesis: