3581 = (2 * 1790) + 1 ;
hence not 2 divides 3581 by NAT_4:9; :: thesis:
3581 = (3 * 1193) + 2 ;
hence not 3 divides 3581 by NAT_4:9; :: thesis:
3581 = (5 * 716) + 1 ;
hence not 5 divides 3581 by NAT_4:9; :: thesis:
3581 = (7 * 511) + 4 ;
hence not 7 divides 3581 by NAT_4:9; :: thesis:
3581 = (11 * 325) + 6 ;
hence not 11 divides 3581 by NAT_4:9; :: thesis:
3581 = (13 * 275) + 6 ;
hence not 13 divides 3581 by NAT_4:9; :: thesis:
3581 = (17 * 210) + 11 ;
hence not 17 divides 3581 by NAT_4:9; :: thesis:
3581 = (19 * 188) + 9 ;
hence not 19 divides 3581 by NAT_4:9; :: thesis:
3581 = (23 * 155) + 16 ;
hence not 23 divides 3581 by NAT_4:9; :: thesis:
3581 = (29 * 123) + 14 ;
hence not 29 divides 3581 by NAT_4:9; :: thesis:
3581 = (31 * 115) + 16 ;
hence not 31 divides 3581 by NAT_4:9; :: thesis:
3581 = (37 * 96) + 29 ;
hence not 37 divides 3581 by NAT_4:9; :: thesis:
3581 = (41 * 87) + 14 ;
hence not 41 divides 3581 by NAT_4:9; :: thesis:
3581 = (43 * 83) + 12 ;
hence not 43 divides 3581 by NAT_4:9; :: thesis:
3581 = (47 * 76) + 9 ;
hence not 47 divides 3581 by NAT_4:9; :: thesis:
3581 = (53 * 67) + 30 ;
hence not 53 divides 3581 by NAT_4:9; :: thesis:
3581 = (59 * 60) + 41 ;
hence not 59 divides 3581 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3581 & n is prime holds
not n divides 3581 by XPRIMET1:34;
hence 3581 is prime by NAT_4:14; :: thesis: