1571 = (2 * 785) + 1 ;
hence not 2 divides 1571 by NAT_4:9; :: thesis:
1571 = (3 * 523) + 2 ;
hence not 3 divides 1571 by NAT_4:9; :: thesis:
1571 = (5 * 314) + 1 ;
hence not 5 divides 1571 by NAT_4:9; :: thesis:
1571 = (7 * 224) + 3 ;
hence not 7 divides 1571 by NAT_4:9; :: thesis:
1571 = (11 * 142) + 9 ;
hence not 11 divides 1571 by NAT_4:9; :: thesis:
1571 = (13 * 120) + 11 ;
hence not 13 divides 1571 by NAT_4:9; :: thesis:
1571 = (17 * 92) + 7 ;
hence not 17 divides 1571 by NAT_4:9; :: thesis:
1571 = (19 * 82) + 13 ;
hence not 19 divides 1571 by NAT_4:9; :: thesis:
1571 = (23 * 68) + 7 ;
hence not 23 divides 1571 by NAT_4:9; :: thesis:
1571 = (29 * 54) + 5 ;
hence not 29 divides 1571 by NAT_4:9; :: thesis:
1571 = (31 * 50) + 21 ;
hence not 31 divides 1571 by NAT_4:9; :: thesis:
1571 = (37 * 42) + 17 ;
hence not 37 divides 1571 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1571 & n is prime holds
not n divides 1571 by XPRIMET1:24;
hence 1571 is prime by NAT_4:14; :: thesis: