1613 = (2 * 806) + 1 ;
hence not 2 divides 1613 by NAT_4:9; :: thesis:
1613 = (3 * 537) + 2 ;
hence not 3 divides 1613 by NAT_4:9; :: thesis:
1613 = (5 * 322) + 3 ;
hence not 5 divides 1613 by NAT_4:9; :: thesis:
1613 = (7 * 230) + 3 ;
hence not 7 divides 1613 by NAT_4:9; :: thesis:
1613 = (11 * 146) + 7 ;
hence not 11 divides 1613 by NAT_4:9; :: thesis:
1613 = (13 * 124) + 1 ;
hence not 13 divides 1613 by NAT_4:9; :: thesis:
1613 = (17 * 94) + 15 ;
hence not 17 divides 1613 by NAT_4:9; :: thesis:
1613 = (19 * 84) + 17 ;
hence not 19 divides 1613 by NAT_4:9; :: thesis:
1613 = (23 * 70) + 3 ;
hence not 23 divides 1613 by NAT_4:9; :: thesis:
1613 = (29 * 55) + 18 ;
hence not 29 divides 1613 by NAT_4:9; :: thesis:
1613 = (31 * 52) + 1 ;
hence not 31 divides 1613 by NAT_4:9; :: thesis:
1613 = (37 * 43) + 22 ;
hence not 37 divides 1613 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1613 & n is prime holds
not n divides 1613 by XPRIMET1:24;
hence 1613 is prime by NAT_4:14; :: thesis: