5689 = (2 * 2844) + 1 ;
hence not 2 divides 5689 by NAT_4:9; :: thesis:
5689 = (3 * 1896) + 1 ;
hence not 3 divides 5689 by NAT_4:9; :: thesis:
5689 = (5 * 1137) + 4 ;
hence not 5 divides 5689 by NAT_4:9; :: thesis:
5689 = (7 * 812) + 5 ;
hence not 7 divides 5689 by NAT_4:9; :: thesis:
5689 = (11 * 517) + 2 ;
hence not 11 divides 5689 by NAT_4:9; :: thesis:
5689 = (13 * 437) + 8 ;
hence not 13 divides 5689 by NAT_4:9; :: thesis:
5689 = (17 * 334) + 11 ;
hence not 17 divides 5689 by NAT_4:9; :: thesis:
5689 = (19 * 299) + 8 ;
hence not 19 divides 5689 by NAT_4:9; :: thesis:
5689 = (23 * 247) + 8 ;
hence not 23 divides 5689 by NAT_4:9; :: thesis:
5689 = (29 * 196) + 5 ;
hence not 29 divides 5689 by NAT_4:9; :: thesis:
5689 = (31 * 183) + 16 ;
hence not 31 divides 5689 by NAT_4:9; :: thesis:
5689 = (37 * 153) + 28 ;
hence not 37 divides 5689 by NAT_4:9; :: thesis:
5689 = (41 * 138) + 31 ;
hence not 41 divides 5689 by NAT_4:9; :: thesis:
5689 = (43 * 132) + 13 ;
hence not 43 divides 5689 by NAT_4:9; :: thesis:
5689 = (47 * 121) + 2 ;
hence not 47 divides 5689 by NAT_4:9; :: thesis:
5689 = (53 * 107) + 18 ;
hence not 53 divides 5689 by NAT_4:9; :: thesis:
5689 = (59 * 96) + 25 ;
hence not 59 divides 5689 by NAT_4:9; :: thesis:
5689 = (61 * 93) + 16 ;
hence not 61 divides 5689 by NAT_4:9; :: thesis:
5689 = (67 * 84) + 61 ;
hence not 67 divides 5689 by NAT_4:9; :: thesis:
5689 = (71 * 80) + 9 ;
hence not 71 divides 5689 by NAT_4:9; :: thesis:
5689 = (73 * 77) + 68 ;
hence not 73 divides 5689 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5689 & n is prime holds
not n divides 5689 by XPRIMET1:42;
hence 5689 is prime by NAT_4:14; :: thesis: