3989 = (2 * 1994) + 1 ;
hence not 2 divides 3989 by NAT_4:9; :: thesis:
3989 = (3 * 1329) + 2 ;
hence not 3 divides 3989 by NAT_4:9; :: thesis:
3989 = (5 * 797) + 4 ;
hence not 5 divides 3989 by NAT_4:9; :: thesis:
3989 = (7 * 569) + 6 ;
hence not 7 divides 3989 by NAT_4:9; :: thesis:
3989 = (11 * 362) + 7 ;
hence not 11 divides 3989 by NAT_4:9; :: thesis:
3989 = (13 * 306) + 11 ;
hence not 13 divides 3989 by NAT_4:9; :: thesis:
3989 = (17 * 234) + 11 ;
hence not 17 divides 3989 by NAT_4:9; :: thesis:
3989 = (19 * 209) + 18 ;
hence not 19 divides 3989 by NAT_4:9; :: thesis:
3989 = (23 * 173) + 10 ;
hence not 23 divides 3989 by NAT_4:9; :: thesis:
3989 = (29 * 137) + 16 ;
hence not 29 divides 3989 by NAT_4:9; :: thesis:
3989 = (31 * 128) + 21 ;
hence not 31 divides 3989 by NAT_4:9; :: thesis:
3989 = (37 * 107) + 30 ;
hence not 37 divides 3989 by NAT_4:9; :: thesis:
3989 = (41 * 97) + 12 ;
hence not 41 divides 3989 by NAT_4:9; :: thesis:
3989 = (43 * 92) + 33 ;
hence not 43 divides 3989 by NAT_4:9; :: thesis:
3989 = (47 * 84) + 41 ;
hence not 47 divides 3989 by NAT_4:9; :: thesis:
3989 = (53 * 75) + 14 ;
hence not 53 divides 3989 by NAT_4:9; :: thesis:
3989 = (59 * 67) + 36 ;
hence not 59 divides 3989 by NAT_4:9; :: thesis:
3989 = (61 * 65) + 24 ;
hence not 61 divides 3989 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3989 & n is prime holds
not n divides 3989 by XPRIMET1:36;
hence 3989 is prime by NAT_4:14; :: thesis: