6841 = (2 * 3420) + 1 ;
hence not 2 divides 6841 by NAT_4:9; :: thesis:
6841 = (3 * 2280) + 1 ;
hence not 3 divides 6841 by NAT_4:9; :: thesis:
6841 = (5 * 1368) + 1 ;
hence not 5 divides 6841 by NAT_4:9; :: thesis:
6841 = (7 * 977) + 2 ;
hence not 7 divides 6841 by NAT_4:9; :: thesis:
6841 = (11 * 621) + 10 ;
hence not 11 divides 6841 by NAT_4:9; :: thesis:
6841 = (13 * 526) + 3 ;
hence not 13 divides 6841 by NAT_4:9; :: thesis:
6841 = (17 * 402) + 7 ;
hence not 17 divides 6841 by NAT_4:9; :: thesis:
6841 = (19 * 360) + 1 ;
hence not 19 divides 6841 by NAT_4:9; :: thesis:
6841 = (23 * 297) + 10 ;
hence not 23 divides 6841 by NAT_4:9; :: thesis:
6841 = (29 * 235) + 26 ;
hence not 29 divides 6841 by NAT_4:9; :: thesis:
6841 = (31 * 220) + 21 ;
hence not 31 divides 6841 by NAT_4:9; :: thesis:
6841 = (37 * 184) + 33 ;
hence not 37 divides 6841 by NAT_4:9; :: thesis:
6841 = (41 * 166) + 35 ;
hence not 41 divides 6841 by NAT_4:9; :: thesis:
6841 = (43 * 159) + 4 ;
hence not 43 divides 6841 by NAT_4:9; :: thesis:
6841 = (47 * 145) + 26 ;
hence not 47 divides 6841 by NAT_4:9; :: thesis:
6841 = (53 * 129) + 4 ;
hence not 53 divides 6841 by NAT_4:9; :: thesis:
6841 = (59 * 115) + 56 ;
hence not 59 divides 6841 by NAT_4:9; :: thesis:
6841 = (61 * 112) + 9 ;
hence not 61 divides 6841 by NAT_4:9; :: thesis:
6841 = (67 * 102) + 7 ;
hence not 67 divides 6841 by NAT_4:9; :: thesis:
6841 = (71 * 96) + 25 ;
hence not 71 divides 6841 by NAT_4:9; :: thesis:
6841 = (73 * 93) + 52 ;
hence not 73 divides 6841 by NAT_4:9; :: thesis:
6841 = (79 * 86) + 47 ;
hence not 79 divides 6841 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6841 & n is prime holds
not n divides 6841 by XPRIMET1:44;
hence 6841 is prime by NAT_4:14; :: thesis: