4079 = (2 * 2039) + 1 ;
hence not 2 divides 4079 by NAT_4:9; :: thesis:
4079 = (3 * 1359) + 2 ;
hence not 3 divides 4079 by NAT_4:9; :: thesis:
4079 = (5 * 815) + 4 ;
hence not 5 divides 4079 by NAT_4:9; :: thesis:
4079 = (7 * 582) + 5 ;
hence not 7 divides 4079 by NAT_4:9; :: thesis:
4079 = (11 * 370) + 9 ;
hence not 11 divides 4079 by NAT_4:9; :: thesis:
4079 = (13 * 313) + 10 ;
hence not 13 divides 4079 by NAT_4:9; :: thesis:
4079 = (17 * 239) + 16 ;
hence not 17 divides 4079 by NAT_4:9; :: thesis:
4079 = (19 * 214) + 13 ;
hence not 19 divides 4079 by NAT_4:9; :: thesis:
4079 = (23 * 177) + 8 ;
hence not 23 divides 4079 by NAT_4:9; :: thesis:
4079 = (29 * 140) + 19 ;
hence not 29 divides 4079 by NAT_4:9; :: thesis:
4079 = (31 * 131) + 18 ;
hence not 31 divides 4079 by NAT_4:9; :: thesis:
4079 = (37 * 110) + 9 ;
hence not 37 divides 4079 by NAT_4:9; :: thesis:
4079 = (41 * 99) + 20 ;
hence not 41 divides 4079 by NAT_4:9; :: thesis:
4079 = (43 * 94) + 37 ;
hence not 43 divides 4079 by NAT_4:9; :: thesis:
4079 = (47 * 86) + 37 ;
hence not 47 divides 4079 by NAT_4:9; :: thesis:
4079 = (53 * 76) + 51 ;
hence not 53 divides 4079 by NAT_4:9; :: thesis:
4079 = (59 * 69) + 8 ;
hence not 59 divides 4079 by NAT_4:9; :: thesis:
4079 = (61 * 66) + 53 ;
hence not 61 divides 4079 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4079 & n is prime holds
not n divides 4079 by XPRIMET1:36;
hence 4079 is prime by NAT_4:14; :: thesis: