2999 = (2 * 1499) + 1 ;
hence not 2 divides 2999 by NAT_4:9; :: thesis:
2999 = (3 * 999) + 2 ;
hence not 3 divides 2999 by NAT_4:9; :: thesis:
2999 = (5 * 599) + 4 ;
hence not 5 divides 2999 by NAT_4:9; :: thesis:
2999 = (7 * 428) + 3 ;
hence not 7 divides 2999 by NAT_4:9; :: thesis:
2999 = (11 * 272) + 7 ;
hence not 11 divides 2999 by NAT_4:9; :: thesis:
2999 = (13 * 230) + 9 ;
hence not 13 divides 2999 by NAT_4:9; :: thesis:
2999 = (17 * 176) + 7 ;
hence not 17 divides 2999 by NAT_4:9; :: thesis:
2999 = (19 * 157) + 16 ;
hence not 19 divides 2999 by NAT_4:9; :: thesis:
2999 = (23 * 130) + 9 ;
hence not 23 divides 2999 by NAT_4:9; :: thesis:
2999 = (29 * 103) + 12 ;
hence not 29 divides 2999 by NAT_4:9; :: thesis:
2999 = (31 * 96) + 23 ;
hence not 31 divides 2999 by NAT_4:9; :: thesis:
2999 = (37 * 81) + 2 ;
hence not 37 divides 2999 by NAT_4:9; :: thesis:
2999 = (41 * 73) + 6 ;
hence not 41 divides 2999 by NAT_4:9; :: thesis:
2999 = (43 * 69) + 32 ;
hence not 43 divides 2999 by NAT_4:9; :: thesis:
2999 = (47 * 63) + 38 ;
hence not 47 divides 2999 by NAT_4:9; :: thesis:
2999 = (53 * 56) + 31 ;
hence not 53 divides 2999 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2999 & n is prime holds
not n divides 2999 by XPRIMET1:32;
hence 2999 is prime by NAT_4:14; :: thesis: