4139 = (2 * 2069) + 1 ;
hence not 2 divides 4139 by NAT_4:9; :: thesis:
4139 = (3 * 1379) + 2 ;
hence not 3 divides 4139 by NAT_4:9; :: thesis:
4139 = (5 * 827) + 4 ;
hence not 5 divides 4139 by NAT_4:9; :: thesis:
4139 = (7 * 591) + 2 ;
hence not 7 divides 4139 by NAT_4:9; :: thesis:
4139 = (11 * 376) + 3 ;
hence not 11 divides 4139 by NAT_4:9; :: thesis:
4139 = (13 * 318) + 5 ;
hence not 13 divides 4139 by NAT_4:9; :: thesis:
4139 = (17 * 243) + 8 ;
hence not 17 divides 4139 by NAT_4:9; :: thesis:
4139 = (19 * 217) + 16 ;
hence not 19 divides 4139 by NAT_4:9; :: thesis:
4139 = (23 * 179) + 22 ;
hence not 23 divides 4139 by NAT_4:9; :: thesis:
4139 = (29 * 142) + 21 ;
hence not 29 divides 4139 by NAT_4:9; :: thesis:
4139 = (31 * 133) + 16 ;
hence not 31 divides 4139 by NAT_4:9; :: thesis:
4139 = (37 * 111) + 32 ;
hence not 37 divides 4139 by NAT_4:9; :: thesis:
4139 = (41 * 100) + 39 ;
hence not 41 divides 4139 by NAT_4:9; :: thesis:
4139 = (43 * 96) + 11 ;
hence not 43 divides 4139 by NAT_4:9; :: thesis:
4139 = (47 * 88) + 3 ;
hence not 47 divides 4139 by NAT_4:9; :: thesis:
4139 = (53 * 78) + 5 ;
hence not 53 divides 4139 by NAT_4:9; :: thesis:
4139 = (59 * 70) + 9 ;
hence not 59 divides 4139 by NAT_4:9; :: thesis:
4139 = (61 * 67) + 52 ;
hence not 61 divides 4139 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4139 & n is prime holds
not n divides 4139 by XPRIMET1:36;
hence 4139 is prime by NAT_4:14; :: thesis: