1409 = (2 * 704) + 1 ;
hence not 2 divides 1409 by NAT_4:9; :: thesis:
1409 = (3 * 469) + 2 ;
hence not 3 divides 1409 by NAT_4:9; :: thesis:
1409 = (5 * 281) + 4 ;
hence not 5 divides 1409 by NAT_4:9; :: thesis:
1409 = (7 * 201) + 2 ;
hence not 7 divides 1409 by NAT_4:9; :: thesis:
1409 = (11 * 128) + 1 ;
hence not 11 divides 1409 by NAT_4:9; :: thesis:
1409 = (13 * 108) + 5 ;
hence not 13 divides 1409 by NAT_4:9; :: thesis:
1409 = (17 * 82) + 15 ;
hence not 17 divides 1409 by NAT_4:9; :: thesis:
1409 = (19 * 74) + 3 ;
hence not 19 divides 1409 by NAT_4:9; :: thesis:
1409 = (23 * 61) + 6 ;
hence not 23 divides 1409 by NAT_4:9; :: thesis:
1409 = (29 * 48) + 17 ;
hence not 29 divides 1409 by NAT_4:9; :: thesis:
1409 = (31 * 45) + 14 ;
hence not 31 divides 1409 by NAT_4:9; :: thesis:
1409 = (37 * 38) + 3 ;
hence not 37 divides 1409 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1409 & n is prime holds
not n divides 1409 by XPRIMET1:24;
hence 1409 is prime by NAT_4:14; :: thesis: