1097 = (2 * 548) + 1 ;
hence not 2 divides 1097 by NAT_4:9; :: thesis:
1097 = (3 * 365) + 2 ;
hence not 3 divides 1097 by NAT_4:9; :: thesis:
1097 = (5 * 219) + 2 ;
hence not 5 divides 1097 by NAT_4:9; :: thesis:
1097 = (7 * 156) + 5 ;
hence not 7 divides 1097 by NAT_4:9; :: thesis:
1097 = (11 * 99) + 8 ;
hence not 11 divides 1097 by NAT_4:9; :: thesis:
1097 = (13 * 84) + 5 ;
hence not 13 divides 1097 by NAT_4:9; :: thesis:
1097 = (17 * 64) + 9 ;
hence not 17 divides 1097 by NAT_4:9; :: thesis:
1097 = (19 * 57) + 14 ;
hence not 19 divides 1097 by NAT_4:9; :: thesis:
1097 = (23 * 47) + 16 ;
hence not 23 divides 1097 by NAT_4:9; :: thesis:
1097 = (29 * 37) + 24 ;
hence not 29 divides 1097 by NAT_4:9; :: thesis:
1097 = (31 * 35) + 12 ;
hence not 31 divides 1097 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1097 & n is prime holds
not n divides 1097 by XPRIMET1:22;
hence 1097 is prime by NAT_4:14; :: thesis: