3011 = (2 * 1505) + 1 ;
hence not 2 divides 3011 by NAT_4:9; :: thesis:
3011 = (3 * 1003) + 2 ;
hence not 3 divides 3011 by NAT_4:9; :: thesis:
3011 = (5 * 602) + 1 ;
hence not 5 divides 3011 by NAT_4:9; :: thesis:
3011 = (7 * 430) + 1 ;
hence not 7 divides 3011 by NAT_4:9; :: thesis:
3011 = (11 * 273) + 8 ;
hence not 11 divides 3011 by NAT_4:9; :: thesis:
3011 = (13 * 231) + 8 ;
hence not 13 divides 3011 by NAT_4:9; :: thesis:
3011 = (17 * 177) + 2 ;
hence not 17 divides 3011 by NAT_4:9; :: thesis:
3011 = (19 * 158) + 9 ;
hence not 19 divides 3011 by NAT_4:9; :: thesis:
3011 = (23 * 130) + 21 ;
hence not 23 divides 3011 by NAT_4:9; :: thesis:
3011 = (29 * 103) + 24 ;
hence not 29 divides 3011 by NAT_4:9; :: thesis:
3011 = (31 * 97) + 4 ;
hence not 31 divides 3011 by NAT_4:9; :: thesis:
3011 = (37 * 81) + 14 ;
hence not 37 divides 3011 by NAT_4:9; :: thesis:
3011 = (41 * 73) + 18 ;
hence not 41 divides 3011 by NAT_4:9; :: thesis:
3011 = (43 * 70) + 1 ;
hence not 43 divides 3011 by NAT_4:9; :: thesis:
3011 = (47 * 64) + 3 ;
hence not 47 divides 3011 by NAT_4:9; :: thesis:
3011 = (53 * 56) + 43 ;
hence not 53 divides 3011 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3011 & n is prime holds
not n divides 3011 by XPRIMET1:32;
hence 3011 is prime by NAT_4:14; :: thesis: