409 = (2 * 204) + 1 ;
hence not 2 divides 409 by NAT_4:9; :: thesis:
409 = (3 * 136) + 1 ;
hence not 3 divides 409 by NAT_4:9; :: thesis:
409 = (5 * 81) + 4 ;
hence not 5 divides 409 by NAT_4:9; :: thesis:
409 = (7 * 58) + 3 ;
hence not 7 divides 409 by NAT_4:9; :: thesis:
409 = (11 * 37) + 2 ;
hence not 11 divides 409 by NAT_4:9; :: thesis:
409 = (13 * 31) + 6 ;
hence not 13 divides 409 by NAT_4:9; :: thesis:
409 = (17 * 24) + 1 ;
hence not 17 divides 409 by NAT_4:9; :: thesis:
409 = (19 * 21) + 10 ;
hence not 19 divides 409 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 409 & n is prime holds
not n divides 409 by XPRIMET1:16;
hence 409 is prime by NAT_4:14; :: thesis: