9689 = (2 * 4844) + 1 ;
hence not 2 divides 9689 by NAT_4:9; :: thesis:
9689 = (3 * 3229) + 2 ;
hence not 3 divides 9689 by NAT_4:9; :: thesis:
9689 = (5 * 1937) + 4 ;
hence not 5 divides 9689 by NAT_4:9; :: thesis:
9689 = (7 * 1384) + 1 ;
hence not 7 divides 9689 by NAT_4:9; :: thesis:
9689 = (11 * 880) + 9 ;
hence not 11 divides 9689 by NAT_4:9; :: thesis:
9689 = (13 * 745) + 4 ;
hence not 13 divides 9689 by NAT_4:9; :: thesis:
9689 = (17 * 569) + 16 ;
hence not 17 divides 9689 by NAT_4:9; :: thesis:
9689 = (19 * 509) + 18 ;
hence not 19 divides 9689 by NAT_4:9; :: thesis:
9689 = (23 * 421) + 6 ;
hence not 23 divides 9689 by NAT_4:9; :: thesis:
9689 = (29 * 334) + 3 ;
hence not 29 divides 9689 by NAT_4:9; :: thesis:
9689 = (31 * 312) + 17 ;
hence not 31 divides 9689 by NAT_4:9; :: thesis:
9689 = (37 * 261) + 32 ;
hence not 37 divides 9689 by NAT_4:9; :: thesis:
9689 = (41 * 236) + 13 ;
hence not 41 divides 9689 by NAT_4:9; :: thesis:
9689 = (43 * 225) + 14 ;
hence not 43 divides 9689 by NAT_4:9; :: thesis:
9689 = (47 * 206) + 7 ;
hence not 47 divides 9689 by NAT_4:9; :: thesis:
9689 = (53 * 182) + 43 ;
hence not 53 divides 9689 by NAT_4:9; :: thesis:
9689 = (59 * 164) + 13 ;
hence not 59 divides 9689 by NAT_4:9; :: thesis:
9689 = (61 * 158) + 51 ;
hence not 61 divides 9689 by NAT_4:9; :: thesis:
9689 = (67 * 144) + 41 ;
hence not 67 divides 9689 by NAT_4:9; :: thesis:
9689 = (71 * 136) + 33 ;
hence not 71 divides 9689 by NAT_4:9; :: thesis:
9689 = (73 * 132) + 53 ;
hence not 73 divides 9689 by NAT_4:9; :: thesis:
9689 = (79 * 122) + 51 ;
hence not 79 divides 9689 by NAT_4:9; :: thesis:
9689 = (83 * 116) + 61 ;
hence not 83 divides 9689 by NAT_4:9; :: thesis:
9689 = (89 * 108) + 77 ;
hence not 89 divides 9689 by NAT_4:9; :: thesis:
9689 = (97 * 99) + 86 ;
hence not 97 divides 9689 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9689 & n is prime holds
not n divides 9689 by XPRIMET1:50;
hence 9689 is prime by NAT_4:14; :: thesis: