3571 = (2 * 1785) + 1 ;
hence not 2 divides 3571 by NAT_4:9; :: thesis:
3571 = (3 * 1190) + 1 ;
hence not 3 divides 3571 by NAT_4:9; :: thesis:
3571 = (5 * 714) + 1 ;
hence not 5 divides 3571 by NAT_4:9; :: thesis:
3571 = (7 * 510) + 1 ;
hence not 7 divides 3571 by NAT_4:9; :: thesis:
3571 = (11 * 324) + 7 ;
hence not 11 divides 3571 by NAT_4:9; :: thesis:
3571 = (13 * 274) + 9 ;
hence not 13 divides 3571 by NAT_4:9; :: thesis:
3571 = (17 * 210) + 1 ;
hence not 17 divides 3571 by NAT_4:9; :: thesis:
3571 = (19 * 187) + 18 ;
hence not 19 divides 3571 by NAT_4:9; :: thesis:
3571 = (23 * 155) + 6 ;
hence not 23 divides 3571 by NAT_4:9; :: thesis:
3571 = (29 * 123) + 4 ;
hence not 29 divides 3571 by NAT_4:9; :: thesis:
3571 = (31 * 115) + 6 ;
hence not 31 divides 3571 by NAT_4:9; :: thesis:
3571 = (37 * 96) + 19 ;
hence not 37 divides 3571 by NAT_4:9; :: thesis:
3571 = (41 * 87) + 4 ;
hence not 41 divides 3571 by NAT_4:9; :: thesis:
3571 = (43 * 83) + 2 ;
hence not 43 divides 3571 by NAT_4:9; :: thesis:
3571 = (47 * 75) + 46 ;
hence not 47 divides 3571 by NAT_4:9; :: thesis:
3571 = (53 * 67) + 20 ;
hence not 53 divides 3571 by NAT_4:9; :: thesis:
3571 = (59 * 60) + 31 ;
hence not 59 divides 3571 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3571 & n is prime holds
not n divides 3571 by XPRIMET1:34;
hence 3571 is prime by NAT_4:14; :: thesis: