5039 = (2 * 2519) + 1 ;
hence not 2 divides 5039 by NAT_4:9; :: thesis:
5039 = (3 * 1679) + 2 ;
hence not 3 divides 5039 by NAT_4:9; :: thesis:
5039 = (5 * 1007) + 4 ;
hence not 5 divides 5039 by NAT_4:9; :: thesis:
5039 = (7 * 719) + 6 ;
hence not 7 divides 5039 by NAT_4:9; :: thesis:
5039 = (11 * 458) + 1 ;
hence not 11 divides 5039 by NAT_4:9; :: thesis:
5039 = (13 * 387) + 8 ;
hence not 13 divides 5039 by NAT_4:9; :: thesis:
5039 = (17 * 296) + 7 ;
hence not 17 divides 5039 by NAT_4:9; :: thesis:
5039 = (19 * 265) + 4 ;
hence not 19 divides 5039 by NAT_4:9; :: thesis:
5039 = (23 * 219) + 2 ;
hence not 23 divides 5039 by NAT_4:9; :: thesis:
5039 = (29 * 173) + 22 ;
hence not 29 divides 5039 by NAT_4:9; :: thesis:
5039 = (31 * 162) + 17 ;
hence not 31 divides 5039 by NAT_4:9; :: thesis:
5039 = (37 * 136) + 7 ;
hence not 37 divides 5039 by NAT_4:9; :: thesis:
5039 = (41 * 122) + 37 ;
hence not 41 divides 5039 by NAT_4:9; :: thesis:
5039 = (43 * 117) + 8 ;
hence not 43 divides 5039 by NAT_4:9; :: thesis:
5039 = (47 * 107) + 10 ;
hence not 47 divides 5039 by NAT_4:9; :: thesis:
5039 = (53 * 95) + 4 ;
hence not 53 divides 5039 by NAT_4:9; :: thesis:
5039 = (59 * 85) + 24 ;
hence not 59 divides 5039 by NAT_4:9; :: thesis:
5039 = (61 * 82) + 37 ;
hence not 61 divides 5039 by NAT_4:9; :: thesis:
5039 = (67 * 75) + 14 ;
hence not 67 divides 5039 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5039 & n is prime holds
not n divides 5039 by XPRIMET1:38;
hence 5039 is prime by NAT_4:14; :: thesis: