1609 = (2 * 804) + 1 ;
hence not 2 divides 1609 by NAT_4:9; :: thesis:
1609 = (3 * 536) + 1 ;
hence not 3 divides 1609 by NAT_4:9; :: thesis:
1609 = (5 * 321) + 4 ;
hence not 5 divides 1609 by NAT_4:9; :: thesis:
1609 = (7 * 229) + 6 ;
hence not 7 divides 1609 by NAT_4:9; :: thesis:
1609 = (11 * 146) + 3 ;
hence not 11 divides 1609 by NAT_4:9; :: thesis:
1609 = (13 * 123) + 10 ;
hence not 13 divides 1609 by NAT_4:9; :: thesis:
1609 = (17 * 94) + 11 ;
hence not 17 divides 1609 by NAT_4:9; :: thesis:
1609 = (19 * 84) + 13 ;
hence not 19 divides 1609 by NAT_4:9; :: thesis:
1609 = (23 * 69) + 22 ;
hence not 23 divides 1609 by NAT_4:9; :: thesis:
1609 = (29 * 55) + 14 ;
hence not 29 divides 1609 by NAT_4:9; :: thesis:
1609 = (31 * 51) + 28 ;
hence not 31 divides 1609 by NAT_4:9; :: thesis:
1609 = (37 * 43) + 18 ;
hence not 37 divides 1609 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1609 & n is prime holds
not n divides 1609 by XPRIMET1:24;
hence 1609 is prime by NAT_4:14; :: thesis: