6143 = (2 * 3071) + 1 ;
hence not 2 divides 6143 by NAT_4:9; :: thesis:
6143 = (3 * 2047) + 2 ;
hence not 3 divides 6143 by NAT_4:9; :: thesis:
6143 = (5 * 1228) + 3 ;
hence not 5 divides 6143 by NAT_4:9; :: thesis:
6143 = (7 * 877) + 4 ;
hence not 7 divides 6143 by NAT_4:9; :: thesis:
6143 = (11 * 558) + 5 ;
hence not 11 divides 6143 by NAT_4:9; :: thesis:
6143 = (13 * 472) + 7 ;
hence not 13 divides 6143 by NAT_4:9; :: thesis:
6143 = (17 * 361) + 6 ;
hence not 17 divides 6143 by NAT_4:9; :: thesis:
6143 = (19 * 323) + 6 ;
hence not 19 divides 6143 by NAT_4:9; :: thesis:
6143 = (23 * 267) + 2 ;
hence not 23 divides 6143 by NAT_4:9; :: thesis:
6143 = (29 * 211) + 24 ;
hence not 29 divides 6143 by NAT_4:9; :: thesis:
6143 = (31 * 198) + 5 ;
hence not 31 divides 6143 by NAT_4:9; :: thesis:
6143 = (37 * 166) + 1 ;
hence not 37 divides 6143 by NAT_4:9; :: thesis:
6143 = (41 * 149) + 34 ;
hence not 41 divides 6143 by NAT_4:9; :: thesis:
6143 = (43 * 142) + 37 ;
hence not 43 divides 6143 by NAT_4:9; :: thesis:
6143 = (47 * 130) + 33 ;
hence not 47 divides 6143 by NAT_4:9; :: thesis:
6143 = (53 * 115) + 48 ;
hence not 53 divides 6143 by NAT_4:9; :: thesis:
6143 = (59 * 104) + 7 ;
hence not 59 divides 6143 by NAT_4:9; :: thesis:
6143 = (61 * 100) + 43 ;
hence not 61 divides 6143 by NAT_4:9; :: thesis:
6143 = (67 * 91) + 46 ;
hence not 67 divides 6143 by NAT_4:9; :: thesis:
6143 = (71 * 86) + 37 ;
hence not 71 divides 6143 by NAT_4:9; :: thesis:
6143 = (73 * 84) + 11 ;
hence not 73 divides 6143 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6143 & n is prime holds
not n divides 6143 by XPRIMET1:42;
hence 6143 is prime by NAT_4:14; :: thesis: