9677 = (2 * 4838) + 1 ;
hence not 2 divides 9677 by NAT_4:9; :: thesis:
9677 = (3 * 3225) + 2 ;
hence not 3 divides 9677 by NAT_4:9; :: thesis:
9677 = (5 * 1935) + 2 ;
hence not 5 divides 9677 by NAT_4:9; :: thesis:
9677 = (7 * 1382) + 3 ;
hence not 7 divides 9677 by NAT_4:9; :: thesis:
9677 = (11 * 879) + 8 ;
hence not 11 divides 9677 by NAT_4:9; :: thesis:
9677 = (13 * 744) + 5 ;
hence not 13 divides 9677 by NAT_4:9; :: thesis:
9677 = (17 * 569) + 4 ;
hence not 17 divides 9677 by NAT_4:9; :: thesis:
9677 = (19 * 509) + 6 ;
hence not 19 divides 9677 by NAT_4:9; :: thesis:
9677 = (23 * 420) + 17 ;
hence not 23 divides 9677 by NAT_4:9; :: thesis:
9677 = (29 * 333) + 20 ;
hence not 29 divides 9677 by NAT_4:9; :: thesis:
9677 = (31 * 312) + 5 ;
hence not 31 divides 9677 by NAT_4:9; :: thesis:
9677 = (37 * 261) + 20 ;
hence not 37 divides 9677 by NAT_4:9; :: thesis:
9677 = (41 * 236) + 1 ;
hence not 41 divides 9677 by NAT_4:9; :: thesis:
9677 = (43 * 225) + 2 ;
hence not 43 divides 9677 by NAT_4:9; :: thesis:
9677 = (47 * 205) + 42 ;
hence not 47 divides 9677 by NAT_4:9; :: thesis:
9677 = (53 * 182) + 31 ;
hence not 53 divides 9677 by NAT_4:9; :: thesis:
9677 = (59 * 164) + 1 ;
hence not 59 divides 9677 by NAT_4:9; :: thesis:
9677 = (61 * 158) + 39 ;
hence not 61 divides 9677 by NAT_4:9; :: thesis:
9677 = (67 * 144) + 29 ;
hence not 67 divides 9677 by NAT_4:9; :: thesis:
9677 = (71 * 136) + 21 ;
hence not 71 divides 9677 by NAT_4:9; :: thesis:
9677 = (73 * 132) + 41 ;
hence not 73 divides 9677 by NAT_4:9; :: thesis:
9677 = (79 * 122) + 39 ;
hence not 79 divides 9677 by NAT_4:9; :: thesis:
9677 = (83 * 116) + 49 ;
hence not 83 divides 9677 by NAT_4:9; :: thesis:
9677 = (89 * 108) + 65 ;
hence not 89 divides 9677 by NAT_4:9; :: thesis:
9677 = (97 * 99) + 74 ;
hence not 97 divides 9677 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9677 & n is prime holds
not n divides 9677 by XPRIMET1:50;
hence 9677 is prime by NAT_4:14; :: thesis: