4349 = (2 * 2174) + 1 ;
hence not 2 divides 4349 by NAT_4:9; :: thesis:
4349 = (3 * 1449) + 2 ;
hence not 3 divides 4349 by NAT_4:9; :: thesis:
4349 = (5 * 869) + 4 ;
hence not 5 divides 4349 by NAT_4:9; :: thesis:
4349 = (7 * 621) + 2 ;
hence not 7 divides 4349 by NAT_4:9; :: thesis:
4349 = (11 * 395) + 4 ;
hence not 11 divides 4349 by NAT_4:9; :: thesis:
4349 = (13 * 334) + 7 ;
hence not 13 divides 4349 by NAT_4:9; :: thesis:
4349 = (17 * 255) + 14 ;
hence not 17 divides 4349 by NAT_4:9; :: thesis:
4349 = (19 * 228) + 17 ;
hence not 19 divides 4349 by NAT_4:9; :: thesis:
4349 = (23 * 189) + 2 ;
hence not 23 divides 4349 by NAT_4:9; :: thesis:
4349 = (29 * 149) + 28 ;
hence not 29 divides 4349 by NAT_4:9; :: thesis:
4349 = (31 * 140) + 9 ;
hence not 31 divides 4349 by NAT_4:9; :: thesis:
4349 = (37 * 117) + 20 ;
hence not 37 divides 4349 by NAT_4:9; :: thesis:
4349 = (41 * 106) + 3 ;
hence not 41 divides 4349 by NAT_4:9; :: thesis:
4349 = (43 * 101) + 6 ;
hence not 43 divides 4349 by NAT_4:9; :: thesis:
4349 = (47 * 92) + 25 ;
hence not 47 divides 4349 by NAT_4:9; :: thesis:
4349 = (53 * 82) + 3 ;
hence not 53 divides 4349 by NAT_4:9; :: thesis:
4349 = (59 * 73) + 42 ;
hence not 59 divides 4349 by NAT_4:9; :: thesis:
4349 = (61 * 71) + 18 ;
hence not 61 divides 4349 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4349 & n is prime holds
not n divides 4349 by XPRIMET1:36;
hence 4349 is prime by NAT_4:14; :: thesis: