1567 = (2 * 783) + 1 ;
hence not 2 divides 1567 by NAT_4:9; :: thesis:
1567 = (3 * 522) + 1 ;
hence not 3 divides 1567 by NAT_4:9; :: thesis:
1567 = (5 * 313) + 2 ;
hence not 5 divides 1567 by NAT_4:9; :: thesis:
1567 = (7 * 223) + 6 ;
hence not 7 divides 1567 by NAT_4:9; :: thesis:
1567 = (11 * 142) + 5 ;
hence not 11 divides 1567 by NAT_4:9; :: thesis:
1567 = (13 * 120) + 7 ;
hence not 13 divides 1567 by NAT_4:9; :: thesis:
1567 = (17 * 92) + 3 ;
hence not 17 divides 1567 by NAT_4:9; :: thesis:
1567 = (19 * 82) + 9 ;
hence not 19 divides 1567 by NAT_4:9; :: thesis:
1567 = (23 * 68) + 3 ;
hence not 23 divides 1567 by NAT_4:9; :: thesis:
1567 = (29 * 54) + 1 ;
hence not 29 divides 1567 by NAT_4:9; :: thesis:
1567 = (31 * 50) + 17 ;
hence not 31 divides 1567 by NAT_4:9; :: thesis:
1567 = (37 * 42) + 13 ;
hence not 37 divides 1567 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1567 & n is prime holds
not n divides 1567 by XPRIMET1:24;
hence 1567 is prime by NAT_4:14; :: thesis: