5557 = (2 * 2778) + 1 ;
hence not 2 divides 5557 by NAT_4:9; :: thesis:
5557 = (3 * 1852) + 1 ;
hence not 3 divides 5557 by NAT_4:9; :: thesis:
5557 = (5 * 1111) + 2 ;
hence not 5 divides 5557 by NAT_4:9; :: thesis:
5557 = (7 * 793) + 6 ;
hence not 7 divides 5557 by NAT_4:9; :: thesis:
5557 = (11 * 505) + 2 ;
hence not 11 divides 5557 by NAT_4:9; :: thesis:
5557 = (13 * 427) + 6 ;
hence not 13 divides 5557 by NAT_4:9; :: thesis:
5557 = (17 * 326) + 15 ;
hence not 17 divides 5557 by NAT_4:9; :: thesis:
5557 = (19 * 292) + 9 ;
hence not 19 divides 5557 by NAT_4:9; :: thesis:
5557 = (23 * 241) + 14 ;
hence not 23 divides 5557 by NAT_4:9; :: thesis:
5557 = (29 * 191) + 18 ;
hence not 29 divides 5557 by NAT_4:9; :: thesis:
5557 = (31 * 179) + 8 ;
hence not 31 divides 5557 by NAT_4:9; :: thesis:
5557 = (37 * 150) + 7 ;
hence not 37 divides 5557 by NAT_4:9; :: thesis:
5557 = (41 * 135) + 22 ;
hence not 41 divides 5557 by NAT_4:9; :: thesis:
5557 = (43 * 129) + 10 ;
hence not 43 divides 5557 by NAT_4:9; :: thesis:
5557 = (47 * 118) + 11 ;
hence not 47 divides 5557 by NAT_4:9; :: thesis:
5557 = (53 * 104) + 45 ;
hence not 53 divides 5557 by NAT_4:9; :: thesis:
5557 = (59 * 94) + 11 ;
hence not 59 divides 5557 by NAT_4:9; :: thesis:
5557 = (61 * 91) + 6 ;
hence not 61 divides 5557 by NAT_4:9; :: thesis:
5557 = (67 * 82) + 63 ;
hence not 67 divides 5557 by NAT_4:9; :: thesis:
5557 = (71 * 78) + 19 ;
hence not 71 divides 5557 by NAT_4:9; :: thesis:
5557 = (73 * 76) + 9 ;
hence not 73 divides 5557 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5557 & n is prime holds
not n divides 5557 by XPRIMET1:42;
hence 5557 is prime by NAT_4:14; :: thesis: