1657 = (2 * 828) + 1 ;
hence not 2 divides 1657 by NAT_4:9; :: thesis:
1657 = (3 * 552) + 1 ;
hence not 3 divides 1657 by NAT_4:9; :: thesis:
1657 = (5 * 331) + 2 ;
hence not 5 divides 1657 by NAT_4:9; :: thesis:
1657 = (7 * 236) + 5 ;
hence not 7 divides 1657 by NAT_4:9; :: thesis:
1657 = (11 * 150) + 7 ;
hence not 11 divides 1657 by NAT_4:9; :: thesis:
1657 = (13 * 127) + 6 ;
hence not 13 divides 1657 by NAT_4:9; :: thesis:
1657 = (17 * 97) + 8 ;
hence not 17 divides 1657 by NAT_4:9; :: thesis:
1657 = (19 * 87) + 4 ;
hence not 19 divides 1657 by NAT_4:9; :: thesis:
1657 = (23 * 72) + 1 ;
hence not 23 divides 1657 by NAT_4:9; :: thesis:
1657 = (29 * 57) + 4 ;
hence not 29 divides 1657 by NAT_4:9; :: thesis:
1657 = (31 * 53) + 14 ;
hence not 31 divides 1657 by NAT_4:9; :: thesis:
1657 = (37 * 44) + 29 ;
hence not 37 divides 1657 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1657 & n is prime holds
not n divides 1657 by XPRIMET1:24;
hence 1657 is prime by NAT_4:14; :: thesis: