8629 = (2 * 4314) + 1 ;
hence not 2 divides 8629 by NAT_4:9; :: thesis:
8629 = (3 * 2876) + 1 ;
hence not 3 divides 8629 by NAT_4:9; :: thesis:
8629 = (5 * 1725) + 4 ;
hence not 5 divides 8629 by NAT_4:9; :: thesis:
8629 = (7 * 1232) + 5 ;
hence not 7 divides 8629 by NAT_4:9; :: thesis:
8629 = (11 * 784) + 5 ;
hence not 11 divides 8629 by NAT_4:9; :: thesis:
8629 = (13 * 663) + 10 ;
hence not 13 divides 8629 by NAT_4:9; :: thesis:
8629 = (17 * 507) + 10 ;
hence not 17 divides 8629 by NAT_4:9; :: thesis:
8629 = (19 * 454) + 3 ;
hence not 19 divides 8629 by NAT_4:9; :: thesis:
8629 = (23 * 375) + 4 ;
hence not 23 divides 8629 by NAT_4:9; :: thesis:
8629 = (29 * 297) + 16 ;
hence not 29 divides 8629 by NAT_4:9; :: thesis:
8629 = (31 * 278) + 11 ;
hence not 31 divides 8629 by NAT_4:9; :: thesis:
8629 = (37 * 233) + 8 ;
hence not 37 divides 8629 by NAT_4:9; :: thesis:
8629 = (41 * 210) + 19 ;
hence not 41 divides 8629 by NAT_4:9; :: thesis:
8629 = (43 * 200) + 29 ;
hence not 43 divides 8629 by NAT_4:9; :: thesis:
8629 = (47 * 183) + 28 ;
hence not 47 divides 8629 by NAT_4:9; :: thesis:
8629 = (53 * 162) + 43 ;
hence not 53 divides 8629 by NAT_4:9; :: thesis:
8629 = (59 * 146) + 15 ;
hence not 59 divides 8629 by NAT_4:9; :: thesis:
8629 = (61 * 141) + 28 ;
hence not 61 divides 8629 by NAT_4:9; :: thesis:
8629 = (67 * 128) + 53 ;
hence not 67 divides 8629 by NAT_4:9; :: thesis:
8629 = (71 * 121) + 38 ;
hence not 71 divides 8629 by NAT_4:9; :: thesis:
8629 = (73 * 118) + 15 ;
hence not 73 divides 8629 by NAT_4:9; :: thesis:
8629 = (79 * 109) + 18 ;
hence not 79 divides 8629 by NAT_4:9; :: thesis:
8629 = (83 * 103) + 80 ;
hence not 83 divides 8629 by NAT_4:9; :: thesis:
8629 = (89 * 96) + 85 ;
hence not 89 divides 8629 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8629 & n is prime holds
not n divides 8629 by XPRIMET1:48;
hence 8629 is prime by NAT_4:14; :: thesis: