5279 = (2 * 2639) + 1 ;
hence not 2 divides 5279 by NAT_4:9; :: thesis:
5279 = (3 * 1759) + 2 ;
hence not 3 divides 5279 by NAT_4:9; :: thesis:
5279 = (5 * 1055) + 4 ;
hence not 5 divides 5279 by NAT_4:9; :: thesis:
5279 = (7 * 754) + 1 ;
hence not 7 divides 5279 by NAT_4:9; :: thesis:
5279 = (11 * 479) + 10 ;
hence not 11 divides 5279 by NAT_4:9; :: thesis:
5279 = (13 * 406) + 1 ;
hence not 13 divides 5279 by NAT_4:9; :: thesis:
5279 = (17 * 310) + 9 ;
hence not 17 divides 5279 by NAT_4:9; :: thesis:
5279 = (19 * 277) + 16 ;
hence not 19 divides 5279 by NAT_4:9; :: thesis:
5279 = (23 * 229) + 12 ;
hence not 23 divides 5279 by NAT_4:9; :: thesis:
5279 = (29 * 182) + 1 ;
hence not 29 divides 5279 by NAT_4:9; :: thesis:
5279 = (31 * 170) + 9 ;
hence not 31 divides 5279 by NAT_4:9; :: thesis:
5279 = (37 * 142) + 25 ;
hence not 37 divides 5279 by NAT_4:9; :: thesis:
5279 = (41 * 128) + 31 ;
hence not 41 divides 5279 by NAT_4:9; :: thesis:
5279 = (43 * 122) + 33 ;
hence not 43 divides 5279 by NAT_4:9; :: thesis:
5279 = (47 * 112) + 15 ;
hence not 47 divides 5279 by NAT_4:9; :: thesis:
5279 = (53 * 99) + 32 ;
hence not 53 divides 5279 by NAT_4:9; :: thesis:
5279 = (59 * 89) + 28 ;
hence not 59 divides 5279 by NAT_4:9; :: thesis:
5279 = (61 * 86) + 33 ;
hence not 61 divides 5279 by NAT_4:9; :: thesis:
5279 = (67 * 78) + 53 ;
hence not 67 divides 5279 by NAT_4:9; :: thesis:
5279 = (71 * 74) + 25 ;
hence not 71 divides 5279 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5279 & n is prime holds
not n divides 5279 by XPRIMET1:40;
hence 5279 is prime by NAT_4:14; :: thesis: