5581 = (2 * 2790) + 1 ;
hence not 2 divides 5581 by NAT_4:9; :: thesis:
5581 = (3 * 1860) + 1 ;
hence not 3 divides 5581 by NAT_4:9; :: thesis:
5581 = (5 * 1116) + 1 ;
hence not 5 divides 5581 by NAT_4:9; :: thesis:
5581 = (7 * 797) + 2 ;
hence not 7 divides 5581 by NAT_4:9; :: thesis:
5581 = (11 * 507) + 4 ;
hence not 11 divides 5581 by NAT_4:9; :: thesis:
5581 = (13 * 429) + 4 ;
hence not 13 divides 5581 by NAT_4:9; :: thesis:
5581 = (17 * 328) + 5 ;
hence not 17 divides 5581 by NAT_4:9; :: thesis:
5581 = (19 * 293) + 14 ;
hence not 19 divides 5581 by NAT_4:9; :: thesis:
5581 = (23 * 242) + 15 ;
hence not 23 divides 5581 by NAT_4:9; :: thesis:
5581 = (29 * 192) + 13 ;
hence not 29 divides 5581 by NAT_4:9; :: thesis:
5581 = (31 * 180) + 1 ;
hence not 31 divides 5581 by NAT_4:9; :: thesis:
5581 = (37 * 150) + 31 ;
hence not 37 divides 5581 by NAT_4:9; :: thesis:
5581 = (41 * 136) + 5 ;
hence not 41 divides 5581 by NAT_4:9; :: thesis:
5581 = (43 * 129) + 34 ;
hence not 43 divides 5581 by NAT_4:9; :: thesis:
5581 = (47 * 118) + 35 ;
hence not 47 divides 5581 by NAT_4:9; :: thesis:
5581 = (53 * 105) + 16 ;
hence not 53 divides 5581 by NAT_4:9; :: thesis:
5581 = (59 * 94) + 35 ;
hence not 59 divides 5581 by NAT_4:9; :: thesis:
5581 = (61 * 91) + 30 ;
hence not 61 divides 5581 by NAT_4:9; :: thesis:
5581 = (67 * 83) + 20 ;
hence not 67 divides 5581 by NAT_4:9; :: thesis:
5581 = (71 * 78) + 43 ;
hence not 71 divides 5581 by NAT_4:9; :: thesis:
5581 = (73 * 76) + 33 ;
hence not 73 divides 5581 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5581 & n is prime holds
not n divides 5581 by XPRIMET1:42;
hence 5581 is prime by NAT_4:14; :: thesis: