557 = (2 * 278) + 1 ;
hence not 2 divides 557 by NAT_4:9; :: thesis:
557 = (3 * 185) + 2 ;
hence not 3 divides 557 by NAT_4:9; :: thesis:
557 = (5 * 111) + 2 ;
hence not 5 divides 557 by NAT_4:9; :: thesis:
557 = (7 * 79) + 4 ;
hence not 7 divides 557 by NAT_4:9; :: thesis:
557 = (11 * 50) + 7 ;
hence not 11 divides 557 by NAT_4:9; :: thesis:
557 = (13 * 42) + 11 ;
hence not 13 divides 557 by NAT_4:9; :: thesis:
557 = (17 * 32) + 13 ;
hence not 17 divides 557 by NAT_4:9; :: thesis:
557 = (19 * 29) + 6 ;
hence not 19 divides 557 by NAT_4:9; :: thesis:
557 = (23 * 24) + 5 ;
hence not 23 divides 557 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 557 & n is prime holds
not n divides 557 by XPRIMET1:18;
hence 557 is prime by NAT_4:14; :: thesis: