4129 = (2 * 2064) + 1 ;
hence not 2 divides 4129 by NAT_4:9; :: thesis:
4129 = (3 * 1376) + 1 ;
hence not 3 divides 4129 by NAT_4:9; :: thesis:
4129 = (5 * 825) + 4 ;
hence not 5 divides 4129 by NAT_4:9; :: thesis:
4129 = (7 * 589) + 6 ;
hence not 7 divides 4129 by NAT_4:9; :: thesis:
4129 = (11 * 375) + 4 ;
hence not 11 divides 4129 by NAT_4:9; :: thesis:
4129 = (13 * 317) + 8 ;
hence not 13 divides 4129 by NAT_4:9; :: thesis:
4129 = (17 * 242) + 15 ;
hence not 17 divides 4129 by NAT_4:9; :: thesis:
4129 = (19 * 217) + 6 ;
hence not 19 divides 4129 by NAT_4:9; :: thesis:
4129 = (23 * 179) + 12 ;
hence not 23 divides 4129 by NAT_4:9; :: thesis:
4129 = (29 * 142) + 11 ;
hence not 29 divides 4129 by NAT_4:9; :: thesis:
4129 = (31 * 133) + 6 ;
hence not 31 divides 4129 by NAT_4:9; :: thesis:
4129 = (37 * 111) + 22 ;
hence not 37 divides 4129 by NAT_4:9; :: thesis:
4129 = (41 * 100) + 29 ;
hence not 41 divides 4129 by NAT_4:9; :: thesis:
4129 = (43 * 96) + 1 ;
hence not 43 divides 4129 by NAT_4:9; :: thesis:
4129 = (47 * 87) + 40 ;
hence not 47 divides 4129 by NAT_4:9; :: thesis:
4129 = (53 * 77) + 48 ;
hence not 53 divides 4129 by NAT_4:9; :: thesis:
4129 = (59 * 69) + 58 ;
hence not 59 divides 4129 by NAT_4:9; :: thesis:
4129 = (61 * 67) + 42 ;
hence not 61 divides 4129 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4129 & n is prime holds
not n divides 4129 by XPRIMET1:36;
hence 4129 is prime by NAT_4:14; :: thesis: