4001 = (2 * 2000) + 1 ;
hence not 2 divides 4001 by NAT_4:9; :: thesis:
4001 = (3 * 1333) + 2 ;
hence not 3 divides 4001 by NAT_4:9; :: thesis:
4001 = (5 * 800) + 1 ;
hence not 5 divides 4001 by NAT_4:9; :: thesis:
4001 = (7 * 571) + 4 ;
hence not 7 divides 4001 by NAT_4:9; :: thesis:
4001 = (11 * 363) + 8 ;
hence not 11 divides 4001 by NAT_4:9; :: thesis:
4001 = (13 * 307) + 10 ;
hence not 13 divides 4001 by NAT_4:9; :: thesis:
4001 = (17 * 235) + 6 ;
hence not 17 divides 4001 by NAT_4:9; :: thesis:
4001 = (19 * 210) + 11 ;
hence not 19 divides 4001 by NAT_4:9; :: thesis:
4001 = (23 * 173) + 22 ;
hence not 23 divides 4001 by NAT_4:9; :: thesis:
4001 = (29 * 137) + 28 ;
hence not 29 divides 4001 by NAT_4:9; :: thesis:
4001 = (31 * 129) + 2 ;
hence not 31 divides 4001 by NAT_4:9; :: thesis:
4001 = (37 * 108) + 5 ;
hence not 37 divides 4001 by NAT_4:9; :: thesis:
4001 = (41 * 97) + 24 ;
hence not 41 divides 4001 by NAT_4:9; :: thesis:
4001 = (43 * 93) + 2 ;
hence not 43 divides 4001 by NAT_4:9; :: thesis:
4001 = (47 * 85) + 6 ;
hence not 47 divides 4001 by NAT_4:9; :: thesis:
4001 = (53 * 75) + 26 ;
hence not 53 divides 4001 by NAT_4:9; :: thesis:
4001 = (59 * 67) + 48 ;
hence not 59 divides 4001 by NAT_4:9; :: thesis:
4001 = (61 * 65) + 36 ;
hence not 61 divides 4001 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4001 & n is prime holds
not n divides 4001 by XPRIMET1:36;
hence 4001 is prime by NAT_4:14; :: thesis: