1129 = (2 * 564) + 1 ;
hence not 2 divides 1129 by NAT_4:9; :: thesis:
1129 = (3 * 376) + 1 ;
hence not 3 divides 1129 by NAT_4:9; :: thesis:
1129 = (5 * 225) + 4 ;
hence not 5 divides 1129 by NAT_4:9; :: thesis:
1129 = (7 * 161) + 2 ;
hence not 7 divides 1129 by NAT_4:9; :: thesis:
1129 = (11 * 102) + 7 ;
hence not 11 divides 1129 by NAT_4:9; :: thesis:
1129 = (13 * 86) + 11 ;
hence not 13 divides 1129 by NAT_4:9; :: thesis:
1129 = (17 * 66) + 7 ;
hence not 17 divides 1129 by NAT_4:9; :: thesis:
1129 = (19 * 59) + 8 ;
hence not 19 divides 1129 by NAT_4:9; :: thesis:
1129 = (23 * 49) + 2 ;
hence not 23 divides 1129 by NAT_4:9; :: thesis:
1129 = (29 * 38) + 27 ;
hence not 29 divides 1129 by NAT_4:9; :: thesis:
1129 = (31 * 36) + 13 ;
hence not 31 divides 1129 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1129 & n is prime holds
not n divides 1129 by XPRIMET1:22;
hence 1129 is prime by NAT_4:14; :: thesis: