4409 = (2 * 2204) + 1 ;
hence not 2 divides 4409 by NAT_4:9; :: thesis:
4409 = (3 * 1469) + 2 ;
hence not 3 divides 4409 by NAT_4:9; :: thesis:
4409 = (5 * 881) + 4 ;
hence not 5 divides 4409 by NAT_4:9; :: thesis:
4409 = (7 * 629) + 6 ;
hence not 7 divides 4409 by NAT_4:9; :: thesis:
4409 = (11 * 400) + 9 ;
hence not 11 divides 4409 by NAT_4:9; :: thesis:
4409 = (13 * 339) + 2 ;
hence not 13 divides 4409 by NAT_4:9; :: thesis:
4409 = (17 * 259) + 6 ;
hence not 17 divides 4409 by NAT_4:9; :: thesis:
4409 = (19 * 232) + 1 ;
hence not 19 divides 4409 by NAT_4:9; :: thesis:
4409 = (23 * 191) + 16 ;
hence not 23 divides 4409 by NAT_4:9; :: thesis:
4409 = (29 * 152) + 1 ;
hence not 29 divides 4409 by NAT_4:9; :: thesis:
4409 = (31 * 142) + 7 ;
hence not 31 divides 4409 by NAT_4:9; :: thesis:
4409 = (37 * 119) + 6 ;
hence not 37 divides 4409 by NAT_4:9; :: thesis:
4409 = (41 * 107) + 22 ;
hence not 41 divides 4409 by NAT_4:9; :: thesis:
4409 = (43 * 102) + 23 ;
hence not 43 divides 4409 by NAT_4:9; :: thesis:
4409 = (47 * 93) + 38 ;
hence not 47 divides 4409 by NAT_4:9; :: thesis:
4409 = (53 * 83) + 10 ;
hence not 53 divides 4409 by NAT_4:9; :: thesis:
4409 = (59 * 74) + 43 ;
hence not 59 divides 4409 by NAT_4:9; :: thesis:
4409 = (61 * 72) + 17 ;
hence not 61 divides 4409 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4409 & n is prime holds
not n divides 4409 by XPRIMET1:36;
hence 4409 is prime by NAT_4:14; :: thesis: