6899 = (2 * 3449) + 1 ;
hence not 2 divides 6899 by NAT_4:9; :: thesis:
6899 = (3 * 2299) + 2 ;
hence not 3 divides 6899 by NAT_4:9; :: thesis:
6899 = (5 * 1379) + 4 ;
hence not 5 divides 6899 by NAT_4:9; :: thesis:
6899 = (7 * 985) + 4 ;
hence not 7 divides 6899 by NAT_4:9; :: thesis:
6899 = (11 * 627) + 2 ;
hence not 11 divides 6899 by NAT_4:9; :: thesis:
6899 = (13 * 530) + 9 ;
hence not 13 divides 6899 by NAT_4:9; :: thesis:
6899 = (17 * 405) + 14 ;
hence not 17 divides 6899 by NAT_4:9; :: thesis:
6899 = (19 * 363) + 2 ;
hence not 19 divides 6899 by NAT_4:9; :: thesis:
6899 = (23 * 299) + 22 ;
hence not 23 divides 6899 by NAT_4:9; :: thesis:
6899 = (29 * 237) + 26 ;
hence not 29 divides 6899 by NAT_4:9; :: thesis:
6899 = (31 * 222) + 17 ;
hence not 31 divides 6899 by NAT_4:9; :: thesis:
6899 = (37 * 186) + 17 ;
hence not 37 divides 6899 by NAT_4:9; :: thesis:
6899 = (41 * 168) + 11 ;
hence not 41 divides 6899 by NAT_4:9; :: thesis:
6899 = (43 * 160) + 19 ;
hence not 43 divides 6899 by NAT_4:9; :: thesis:
6899 = (47 * 146) + 37 ;
hence not 47 divides 6899 by NAT_4:9; :: thesis:
6899 = (53 * 130) + 9 ;
hence not 53 divides 6899 by NAT_4:9; :: thesis:
6899 = (59 * 116) + 55 ;
hence not 59 divides 6899 by NAT_4:9; :: thesis:
6899 = (61 * 113) + 6 ;
hence not 61 divides 6899 by NAT_4:9; :: thesis:
6899 = (67 * 102) + 65 ;
hence not 67 divides 6899 by NAT_4:9; :: thesis:
6899 = (71 * 97) + 12 ;
hence not 71 divides 6899 by NAT_4:9; :: thesis:
6899 = (73 * 94) + 37 ;
hence not 73 divides 6899 by NAT_4:9; :: thesis:
6899 = (79 * 87) + 26 ;
hence not 79 divides 6899 by NAT_4:9; :: thesis:
6899 = (83 * 83) + 10 ;
hence not 83 divides 6899 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6899 & n is prime holds
not n divides 6899 by XPRIMET1:46;
hence 6899 is prime by NAT_4:14; :: thesis: