5657 = (2 * 2828) + 1 ;
hence not 2 divides 5657 by NAT_4:9; :: thesis:
5657 = (3 * 1885) + 2 ;
hence not 3 divides 5657 by NAT_4:9; :: thesis:
5657 = (5 * 1131) + 2 ;
hence not 5 divides 5657 by NAT_4:9; :: thesis:
5657 = (7 * 808) + 1 ;
hence not 7 divides 5657 by NAT_4:9; :: thesis:
5657 = (11 * 514) + 3 ;
hence not 11 divides 5657 by NAT_4:9; :: thesis:
5657 = (13 * 435) + 2 ;
hence not 13 divides 5657 by NAT_4:9; :: thesis:
5657 = (17 * 332) + 13 ;
hence not 17 divides 5657 by NAT_4:9; :: thesis:
5657 = (19 * 297) + 14 ;
hence not 19 divides 5657 by NAT_4:9; :: thesis:
5657 = (23 * 245) + 22 ;
hence not 23 divides 5657 by NAT_4:9; :: thesis:
5657 = (29 * 195) + 2 ;
hence not 29 divides 5657 by NAT_4:9; :: thesis:
5657 = (31 * 182) + 15 ;
hence not 31 divides 5657 by NAT_4:9; :: thesis:
5657 = (37 * 152) + 33 ;
hence not 37 divides 5657 by NAT_4:9; :: thesis:
5657 = (41 * 137) + 40 ;
hence not 41 divides 5657 by NAT_4:9; :: thesis:
5657 = (43 * 131) + 24 ;
hence not 43 divides 5657 by NAT_4:9; :: thesis:
5657 = (47 * 120) + 17 ;
hence not 47 divides 5657 by NAT_4:9; :: thesis:
5657 = (53 * 106) + 39 ;
hence not 53 divides 5657 by NAT_4:9; :: thesis:
5657 = (59 * 95) + 52 ;
hence not 59 divides 5657 by NAT_4:9; :: thesis:
5657 = (61 * 92) + 45 ;
hence not 61 divides 5657 by NAT_4:9; :: thesis:
5657 = (67 * 84) + 29 ;
hence not 67 divides 5657 by NAT_4:9; :: thesis:
5657 = (71 * 79) + 48 ;
hence not 71 divides 5657 by NAT_4:9; :: thesis:
5657 = (73 * 77) + 36 ;
hence not 73 divides 5657 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5657 & n is prime holds
not n divides 5657 by XPRIMET1:42;
hence 5657 is prime by NAT_4:14; :: thesis: