5309 = (2 * 2654) + 1 ;
hence not 2 divides 5309 by NAT_4:9; :: thesis:
5309 = (3 * 1769) + 2 ;
hence not 3 divides 5309 by NAT_4:9; :: thesis:
5309 = (5 * 1061) + 4 ;
hence not 5 divides 5309 by NAT_4:9; :: thesis:
5309 = (7 * 758) + 3 ;
hence not 7 divides 5309 by NAT_4:9; :: thesis:
5309 = (11 * 482) + 7 ;
hence not 11 divides 5309 by NAT_4:9; :: thesis:
5309 = (13 * 408) + 5 ;
hence not 13 divides 5309 by NAT_4:9; :: thesis:
5309 = (17 * 312) + 5 ;
hence not 17 divides 5309 by NAT_4:9; :: thesis:
5309 = (19 * 279) + 8 ;
hence not 19 divides 5309 by NAT_4:9; :: thesis:
5309 = (23 * 230) + 19 ;
hence not 23 divides 5309 by NAT_4:9; :: thesis:
5309 = (29 * 183) + 2 ;
hence not 29 divides 5309 by NAT_4:9; :: thesis:
5309 = (31 * 171) + 8 ;
hence not 31 divides 5309 by NAT_4:9; :: thesis:
5309 = (37 * 143) + 18 ;
hence not 37 divides 5309 by NAT_4:9; :: thesis:
5309 = (41 * 129) + 20 ;
hence not 41 divides 5309 by NAT_4:9; :: thesis:
5309 = (43 * 123) + 20 ;
hence not 43 divides 5309 by NAT_4:9; :: thesis:
5309 = (47 * 112) + 45 ;
hence not 47 divides 5309 by NAT_4:9; :: thesis:
5309 = (53 * 100) + 9 ;
hence not 53 divides 5309 by NAT_4:9; :: thesis:
5309 = (59 * 89) + 58 ;
hence not 59 divides 5309 by NAT_4:9; :: thesis:
5309 = (61 * 87) + 2 ;
hence not 61 divides 5309 by NAT_4:9; :: thesis:
5309 = (67 * 79) + 16 ;
hence not 67 divides 5309 by NAT_4:9; :: thesis:
5309 = (71 * 74) + 55 ;
hence not 71 divides 5309 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5309 & n is prime holds
not n divides 5309 by XPRIMET1:40;
hence 5309 is prime by NAT_4:14; :: thesis: