1123 = (2 * 561) + 1 ;
hence not 2 divides 1123 by NAT_4:9; :: thesis:
1123 = (3 * 374) + 1 ;
hence not 3 divides 1123 by NAT_4:9; :: thesis:
1123 = (5 * 224) + 3 ;
hence not 5 divides 1123 by NAT_4:9; :: thesis:
1123 = (7 * 160) + 3 ;
hence not 7 divides 1123 by NAT_4:9; :: thesis:
1123 = (11 * 102) + 1 ;
hence not 11 divides 1123 by NAT_4:9; :: thesis:
1123 = (13 * 86) + 5 ;
hence not 13 divides 1123 by NAT_4:9; :: thesis:
1123 = (17 * 66) + 1 ;
hence not 17 divides 1123 by NAT_4:9; :: thesis:
1123 = (19 * 59) + 2 ;
hence not 19 divides 1123 by NAT_4:9; :: thesis:
1123 = (23 * 48) + 19 ;
hence not 23 divides 1123 by NAT_4:9; :: thesis:
1123 = (29 * 38) + 21 ;
hence not 29 divides 1123 by NAT_4:9; :: thesis:
1123 = (31 * 36) + 7 ;
hence not 31 divides 1123 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1123 & n is prime holds
not n divides 1123 by XPRIMET1:22;
hence 1123 is prime by NAT_4:14; :: thesis: