8039 = (2 * 4019) + 1 ;
hence not 2 divides 8039 by NAT_4:9; :: thesis:
8039 = (3 * 2679) + 2 ;
hence not 3 divides 8039 by NAT_4:9; :: thesis:
8039 = (5 * 1607) + 4 ;
hence not 5 divides 8039 by NAT_4:9; :: thesis:
8039 = (7 * 1148) + 3 ;
hence not 7 divides 8039 by NAT_4:9; :: thesis:
8039 = (11 * 730) + 9 ;
hence not 11 divides 8039 by NAT_4:9; :: thesis:
8039 = (13 * 618) + 5 ;
hence not 13 divides 8039 by NAT_4:9; :: thesis:
8039 = (17 * 472) + 15 ;
hence not 17 divides 8039 by NAT_4:9; :: thesis:
8039 = (19 * 423) + 2 ;
hence not 19 divides 8039 by NAT_4:9; :: thesis:
8039 = (23 * 349) + 12 ;
hence not 23 divides 8039 by NAT_4:9; :: thesis:
8039 = (29 * 277) + 6 ;
hence not 29 divides 8039 by NAT_4:9; :: thesis:
8039 = (31 * 259) + 10 ;
hence not 31 divides 8039 by NAT_4:9; :: thesis:
8039 = (37 * 217) + 10 ;
hence not 37 divides 8039 by NAT_4:9; :: thesis:
8039 = (41 * 196) + 3 ;
hence not 41 divides 8039 by NAT_4:9; :: thesis:
8039 = (43 * 186) + 41 ;
hence not 43 divides 8039 by NAT_4:9; :: thesis:
8039 = (47 * 171) + 2 ;
hence not 47 divides 8039 by NAT_4:9; :: thesis:
8039 = (53 * 151) + 36 ;
hence not 53 divides 8039 by NAT_4:9; :: thesis:
8039 = (59 * 136) + 15 ;
hence not 59 divides 8039 by NAT_4:9; :: thesis:
8039 = (61 * 131) + 48 ;
hence not 61 divides 8039 by NAT_4:9; :: thesis:
8039 = (67 * 119) + 66 ;
hence not 67 divides 8039 by NAT_4:9; :: thesis:
8039 = (71 * 113) + 16 ;
hence not 71 divides 8039 by NAT_4:9; :: thesis:
8039 = (73 * 110) + 9 ;
hence not 73 divides 8039 by NAT_4:9; :: thesis:
8039 = (79 * 101) + 60 ;
hence not 79 divides 8039 by NAT_4:9; :: thesis:
8039 = (83 * 96) + 71 ;
hence not 83 divides 8039 by NAT_4:9; :: thesis:
8039 = (89 * 90) + 29 ;
hence not 89 divides 8039 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8039 & n is prime holds
not n divides 8039 by XPRIMET1:48;
hence 8039 is prime by NAT_4:14; :: thesis: