2557 = (2 * 1278) + 1 ;
hence not 2 divides 2557 by NAT_4:9; :: thesis:
2557 = (3 * 852) + 1 ;
hence not 3 divides 2557 by NAT_4:9; :: thesis:
2557 = (5 * 511) + 2 ;
hence not 5 divides 2557 by NAT_4:9; :: thesis:
2557 = (7 * 365) + 2 ;
hence not 7 divides 2557 by NAT_4:9; :: thesis:
2557 = (11 * 232) + 5 ;
hence not 11 divides 2557 by NAT_4:9; :: thesis:
2557 = (13 * 196) + 9 ;
hence not 13 divides 2557 by NAT_4:9; :: thesis:
2557 = (17 * 150) + 7 ;
hence not 17 divides 2557 by NAT_4:9; :: thesis:
2557 = (19 * 134) + 11 ;
hence not 19 divides 2557 by NAT_4:9; :: thesis:
2557 = (23 * 111) + 4 ;
hence not 23 divides 2557 by NAT_4:9; :: thesis:
2557 = (29 * 88) + 5 ;
hence not 29 divides 2557 by NAT_4:9; :: thesis:
2557 = (31 * 82) + 15 ;
hence not 31 divides 2557 by NAT_4:9; :: thesis:
2557 = (37 * 69) + 4 ;
hence not 37 divides 2557 by NAT_4:9; :: thesis:
2557 = (41 * 62) + 15 ;
hence not 41 divides 2557 by NAT_4:9; :: thesis:
2557 = (43 * 59) + 20 ;
hence not 43 divides 2557 by NAT_4:9; :: thesis:
2557 = (47 * 54) + 19 ;
hence not 47 divides 2557 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2557 & n is prime holds
not n divides 2557 by XPRIMET1:30;
hence 2557 is prime by NAT_4:14; :: thesis: