2161 = (2 * 1080) + 1 ;
hence not 2 divides 2161 by NAT_4:9; :: thesis:
2161 = (3 * 720) + 1 ;
hence not 3 divides 2161 by NAT_4:9; :: thesis:
2161 = (5 * 432) + 1 ;
hence not 5 divides 2161 by NAT_4:9; :: thesis:
2161 = (7 * 308) + 5 ;
hence not 7 divides 2161 by NAT_4:9; :: thesis:
2161 = (11 * 196) + 5 ;
hence not 11 divides 2161 by NAT_4:9; :: thesis:
2161 = (13 * 166) + 3 ;
hence not 13 divides 2161 by NAT_4:9; :: thesis:
2161 = (17 * 127) + 2 ;
hence not 17 divides 2161 by NAT_4:9; :: thesis:
2161 = (19 * 113) + 14 ;
hence not 19 divides 2161 by NAT_4:9; :: thesis:
2161 = (23 * 93) + 22 ;
hence not 23 divides 2161 by NAT_4:9; :: thesis:
2161 = (29 * 74) + 15 ;
hence not 29 divides 2161 by NAT_4:9; :: thesis:
2161 = (31 * 69) + 22 ;
hence not 31 divides 2161 by NAT_4:9; :: thesis:
2161 = (37 * 58) + 15 ;
hence not 37 divides 2161 by NAT_4:9; :: thesis:
2161 = (41 * 52) + 29 ;
hence not 41 divides 2161 by NAT_4:9; :: thesis:
2161 = (43 * 50) + 11 ;
hence not 43 divides 2161 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2161 & n is prime holds
not n divides 2161 by XPRIMET1:28;
hence 2161 is prime by NAT_4:14; :: thesis: