6343 = (2 * 3171) + 1 ;
hence not 2 divides 6343 by NAT_4:9; :: thesis:
6343 = (3 * 2114) + 1 ;
hence not 3 divides 6343 by NAT_4:9; :: thesis:
6343 = (5 * 1268) + 3 ;
hence not 5 divides 6343 by NAT_4:9; :: thesis:
6343 = (7 * 906) + 1 ;
hence not 7 divides 6343 by NAT_4:9; :: thesis:
6343 = (11 * 576) + 7 ;
hence not 11 divides 6343 by NAT_4:9; :: thesis:
6343 = (13 * 487) + 12 ;
hence not 13 divides 6343 by NAT_4:9; :: thesis:
6343 = (17 * 373) + 2 ;
hence not 17 divides 6343 by NAT_4:9; :: thesis:
6343 = (19 * 333) + 16 ;
hence not 19 divides 6343 by NAT_4:9; :: thesis:
6343 = (23 * 275) + 18 ;
hence not 23 divides 6343 by NAT_4:9; :: thesis:
6343 = (29 * 218) + 21 ;
hence not 29 divides 6343 by NAT_4:9; :: thesis:
6343 = (31 * 204) + 19 ;
hence not 31 divides 6343 by NAT_4:9; :: thesis:
6343 = (37 * 171) + 16 ;
hence not 37 divides 6343 by NAT_4:9; :: thesis:
6343 = (41 * 154) + 29 ;
hence not 41 divides 6343 by NAT_4:9; :: thesis:
6343 = (43 * 147) + 22 ;
hence not 43 divides 6343 by NAT_4:9; :: thesis:
6343 = (47 * 134) + 45 ;
hence not 47 divides 6343 by NAT_4:9; :: thesis:
6343 = (53 * 119) + 36 ;
hence not 53 divides 6343 by NAT_4:9; :: thesis:
6343 = (59 * 107) + 30 ;
hence not 59 divides 6343 by NAT_4:9; :: thesis:
6343 = (61 * 103) + 60 ;
hence not 61 divides 6343 by NAT_4:9; :: thesis:
6343 = (67 * 94) + 45 ;
hence not 67 divides 6343 by NAT_4:9; :: thesis:
6343 = (71 * 89) + 24 ;
hence not 71 divides 6343 by NAT_4:9; :: thesis:
6343 = (73 * 86) + 65 ;
hence not 73 divides 6343 by NAT_4:9; :: thesis:
6343 = (79 * 80) + 23 ;
hence not 79 divides 6343 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6343 & n is prime holds
not n divides 6343 by XPRIMET1:44;
hence 6343 is prime by NAT_4:14; :: thesis: