4339 = (2 * 2169) + 1 ;
hence not 2 divides 4339 by NAT_4:9; :: thesis:
4339 = (3 * 1446) + 1 ;
hence not 3 divides 4339 by NAT_4:9; :: thesis:
4339 = (5 * 867) + 4 ;
hence not 5 divides 4339 by NAT_4:9; :: thesis:
4339 = (7 * 619) + 6 ;
hence not 7 divides 4339 by NAT_4:9; :: thesis:
4339 = (11 * 394) + 5 ;
hence not 11 divides 4339 by NAT_4:9; :: thesis:
4339 = (13 * 333) + 10 ;
hence not 13 divides 4339 by NAT_4:9; :: thesis:
4339 = (17 * 255) + 4 ;
hence not 17 divides 4339 by NAT_4:9; :: thesis:
4339 = (19 * 228) + 7 ;
hence not 19 divides 4339 by NAT_4:9; :: thesis:
4339 = (23 * 188) + 15 ;
hence not 23 divides 4339 by NAT_4:9; :: thesis:
4339 = (29 * 149) + 18 ;
hence not 29 divides 4339 by NAT_4:9; :: thesis:
4339 = (31 * 139) + 30 ;
hence not 31 divides 4339 by NAT_4:9; :: thesis:
4339 = (37 * 117) + 10 ;
hence not 37 divides 4339 by NAT_4:9; :: thesis:
4339 = (41 * 105) + 34 ;
hence not 41 divides 4339 by NAT_4:9; :: thesis:
4339 = (43 * 100) + 39 ;
hence not 43 divides 4339 by NAT_4:9; :: thesis:
4339 = (47 * 92) + 15 ;
hence not 47 divides 4339 by NAT_4:9; :: thesis:
4339 = (53 * 81) + 46 ;
hence not 53 divides 4339 by NAT_4:9; :: thesis:
4339 = (59 * 73) + 32 ;
hence not 59 divides 4339 by NAT_4:9; :: thesis:
4339 = (61 * 71) + 8 ;
hence not 61 divides 4339 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4339 & n is prime holds
not n divides 4339 by XPRIMET1:36;
hence 4339 is prime by NAT_4:14; :: thesis: