1619 = (2 * 809) + 1 ;
hence not 2 divides 1619 by NAT_4:9; :: thesis:
1619 = (3 * 539) + 2 ;
hence not 3 divides 1619 by NAT_4:9; :: thesis:
1619 = (5 * 323) + 4 ;
hence not 5 divides 1619 by NAT_4:9; :: thesis:
1619 = (7 * 231) + 2 ;
hence not 7 divides 1619 by NAT_4:9; :: thesis:
1619 = (11 * 147) + 2 ;
hence not 11 divides 1619 by NAT_4:9; :: thesis:
1619 = (13 * 124) + 7 ;
hence not 13 divides 1619 by NAT_4:9; :: thesis:
1619 = (17 * 95) + 4 ;
hence not 17 divides 1619 by NAT_4:9; :: thesis:
1619 = (19 * 85) + 4 ;
hence not 19 divides 1619 by NAT_4:9; :: thesis:
1619 = (23 * 70) + 9 ;
hence not 23 divides 1619 by NAT_4:9; :: thesis:
1619 = (29 * 55) + 24 ;
hence not 29 divides 1619 by NAT_4:9; :: thesis:
1619 = (31 * 52) + 7 ;
hence not 31 divides 1619 by NAT_4:9; :: thesis:
1619 = (37 * 43) + 28 ;
hence not 37 divides 1619 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1619 & n is prime holds
not n divides 1619 by XPRIMET1:24;
hence 1619 is prime by NAT_4:14; :: thesis: