3709 = (2 * 1854) + 1 ;
hence not 2 divides 3709 by NAT_4:9; :: thesis:
3709 = (3 * 1236) + 1 ;
hence not 3 divides 3709 by NAT_4:9; :: thesis:
3709 = (5 * 741) + 4 ;
hence not 5 divides 3709 by NAT_4:9; :: thesis:
3709 = (7 * 529) + 6 ;
hence not 7 divides 3709 by NAT_4:9; :: thesis:
3709 = (11 * 337) + 2 ;
hence not 11 divides 3709 by NAT_4:9; :: thesis:
3709 = (13 * 285) + 4 ;
hence not 13 divides 3709 by NAT_4:9; :: thesis:
3709 = (17 * 218) + 3 ;
hence not 17 divides 3709 by NAT_4:9; :: thesis:
3709 = (19 * 195) + 4 ;
hence not 19 divides 3709 by NAT_4:9; :: thesis:
3709 = (23 * 161) + 6 ;
hence not 23 divides 3709 by NAT_4:9; :: thesis:
3709 = (29 * 127) + 26 ;
hence not 29 divides 3709 by NAT_4:9; :: thesis:
3709 = (31 * 119) + 20 ;
hence not 31 divides 3709 by NAT_4:9; :: thesis:
3709 = (37 * 100) + 9 ;
hence not 37 divides 3709 by NAT_4:9; :: thesis:
3709 = (41 * 90) + 19 ;
hence not 41 divides 3709 by NAT_4:9; :: thesis:
3709 = (43 * 86) + 11 ;
hence not 43 divides 3709 by NAT_4:9; :: thesis:
3709 = (47 * 78) + 43 ;
hence not 47 divides 3709 by NAT_4:9; :: thesis:
3709 = (53 * 69) + 52 ;
hence not 53 divides 3709 by NAT_4:9; :: thesis:
3709 = (59 * 62) + 51 ;
hence not 59 divides 3709 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3709 & n is prime holds
not n divides 3709 by XPRIMET1:34;
hence 3709 is prime by NAT_4:14; :: thesis: