6089 = (2 * 3044) + 1 ;
hence not 2 divides 6089 by NAT_4:9; :: thesis:
6089 = (3 * 2029) + 2 ;
hence not 3 divides 6089 by NAT_4:9; :: thesis:
6089 = (5 * 1217) + 4 ;
hence not 5 divides 6089 by NAT_4:9; :: thesis:
6089 = (7 * 869) + 6 ;
hence not 7 divides 6089 by NAT_4:9; :: thesis:
6089 = (11 * 553) + 6 ;
hence not 11 divides 6089 by NAT_4:9; :: thesis:
6089 = (13 * 468) + 5 ;
hence not 13 divides 6089 by NAT_4:9; :: thesis:
6089 = (17 * 358) + 3 ;
hence not 17 divides 6089 by NAT_4:9; :: thesis:
6089 = (19 * 320) + 9 ;
hence not 19 divides 6089 by NAT_4:9; :: thesis:
6089 = (23 * 264) + 17 ;
hence not 23 divides 6089 by NAT_4:9; :: thesis:
6089 = (29 * 209) + 28 ;
hence not 29 divides 6089 by NAT_4:9; :: thesis:
6089 = (31 * 196) + 13 ;
hence not 31 divides 6089 by NAT_4:9; :: thesis:
6089 = (37 * 164) + 21 ;
hence not 37 divides 6089 by NAT_4:9; :: thesis:
6089 = (41 * 148) + 21 ;
hence not 41 divides 6089 by NAT_4:9; :: thesis:
6089 = (43 * 141) + 26 ;
hence not 43 divides 6089 by NAT_4:9; :: thesis:
6089 = (47 * 129) + 26 ;
hence not 47 divides 6089 by NAT_4:9; :: thesis:
6089 = (53 * 114) + 47 ;
hence not 53 divides 6089 by NAT_4:9; :: thesis:
6089 = (59 * 103) + 12 ;
hence not 59 divides 6089 by NAT_4:9; :: thesis:
6089 = (61 * 99) + 50 ;
hence not 61 divides 6089 by NAT_4:9; :: thesis:
6089 = (67 * 90) + 59 ;
hence not 67 divides 6089 by NAT_4:9; :: thesis:
6089 = (71 * 85) + 54 ;
hence not 71 divides 6089 by NAT_4:9; :: thesis:
6089 = (73 * 83) + 30 ;
hence not 73 divides 6089 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6089 & n is prime holds
not n divides 6089 by XPRIMET1:42;
hence 6089 is prime by NAT_4:14; :: thesis: