6869 = (2 * 3434) + 1 ;
hence not 2 divides 6869 by NAT_4:9; :: thesis:
6869 = (3 * 2289) + 2 ;
hence not 3 divides 6869 by NAT_4:9; :: thesis:
6869 = (5 * 1373) + 4 ;
hence not 5 divides 6869 by NAT_4:9; :: thesis:
6869 = (7 * 981) + 2 ;
hence not 7 divides 6869 by NAT_4:9; :: thesis:
6869 = (11 * 624) + 5 ;
hence not 11 divides 6869 by NAT_4:9; :: thesis:
6869 = (13 * 528) + 5 ;
hence not 13 divides 6869 by NAT_4:9; :: thesis:
6869 = (17 * 404) + 1 ;
hence not 17 divides 6869 by NAT_4:9; :: thesis:
6869 = (19 * 361) + 10 ;
hence not 19 divides 6869 by NAT_4:9; :: thesis:
6869 = (23 * 298) + 15 ;
hence not 23 divides 6869 by NAT_4:9; :: thesis:
6869 = (29 * 236) + 25 ;
hence not 29 divides 6869 by NAT_4:9; :: thesis:
6869 = (31 * 221) + 18 ;
hence not 31 divides 6869 by NAT_4:9; :: thesis:
6869 = (37 * 185) + 24 ;
hence not 37 divides 6869 by NAT_4:9; :: thesis:
6869 = (41 * 167) + 22 ;
hence not 41 divides 6869 by NAT_4:9; :: thesis:
6869 = (43 * 159) + 32 ;
hence not 43 divides 6869 by NAT_4:9; :: thesis:
6869 = (47 * 146) + 7 ;
hence not 47 divides 6869 by NAT_4:9; :: thesis:
6869 = (53 * 129) + 32 ;
hence not 53 divides 6869 by NAT_4:9; :: thesis:
6869 = (59 * 116) + 25 ;
hence not 59 divides 6869 by NAT_4:9; :: thesis:
6869 = (61 * 112) + 37 ;
hence not 61 divides 6869 by NAT_4:9; :: thesis:
6869 = (67 * 102) + 35 ;
hence not 67 divides 6869 by NAT_4:9; :: thesis:
6869 = (71 * 96) + 53 ;
hence not 71 divides 6869 by NAT_4:9; :: thesis:
6869 = (73 * 94) + 7 ;
hence not 73 divides 6869 by NAT_4:9; :: thesis:
6869 = (79 * 86) + 75 ;
hence not 79 divides 6869 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6869 & n is prime holds
not n divides 6869 by XPRIMET1:44;
hence 6869 is prime by NAT_4:14; :: thesis: