3407 = (2 * 1703) + 1 ;
hence not 2 divides 3407 by NAT_4:9; :: thesis:
3407 = (3 * 1135) + 2 ;
hence not 3 divides 3407 by NAT_4:9; :: thesis:
3407 = (5 * 681) + 2 ;
hence not 5 divides 3407 by NAT_4:9; :: thesis:
3407 = (7 * 486) + 5 ;
hence not 7 divides 3407 by NAT_4:9; :: thesis:
3407 = (11 * 309) + 8 ;
hence not 11 divides 3407 by NAT_4:9; :: thesis:
3407 = (13 * 262) + 1 ;
hence not 13 divides 3407 by NAT_4:9; :: thesis:
3407 = (17 * 200) + 7 ;
hence not 17 divides 3407 by NAT_4:9; :: thesis:
3407 = (19 * 179) + 6 ;
hence not 19 divides 3407 by NAT_4:9; :: thesis:
3407 = (23 * 148) + 3 ;
hence not 23 divides 3407 by NAT_4:9; :: thesis:
3407 = (29 * 117) + 14 ;
hence not 29 divides 3407 by NAT_4:9; :: thesis:
3407 = (31 * 109) + 28 ;
hence not 31 divides 3407 by NAT_4:9; :: thesis:
3407 = (37 * 92) + 3 ;
hence not 37 divides 3407 by NAT_4:9; :: thesis:
3407 = (41 * 83) + 4 ;
hence not 41 divides 3407 by NAT_4:9; :: thesis:
3407 = (43 * 79) + 10 ;
hence not 43 divides 3407 by NAT_4:9; :: thesis:
3407 = (47 * 72) + 23 ;
hence not 47 divides 3407 by NAT_4:9; :: thesis:
3407 = (53 * 64) + 15 ;
hence not 53 divides 3407 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3407 & n is prime holds
not n divides 3407 by XPRIMET1:32;
hence 3407 is prime by NAT_4:14; :: thesis: