7841 = (2 * 3920) + 1 ;
hence not 2 divides 7841 by NAT_4:9; :: thesis:
7841 = (3 * 2613) + 2 ;
hence not 3 divides 7841 by NAT_4:9; :: thesis:
7841 = (5 * 1568) + 1 ;
hence not 5 divides 7841 by NAT_4:9; :: thesis:
7841 = (7 * 1120) + 1 ;
hence not 7 divides 7841 by NAT_4:9; :: thesis:
7841 = (11 * 712) + 9 ;
hence not 11 divides 7841 by NAT_4:9; :: thesis:
7841 = (13 * 603) + 2 ;
hence not 13 divides 7841 by NAT_4:9; :: thesis:
7841 = (17 * 461) + 4 ;
hence not 17 divides 7841 by NAT_4:9; :: thesis:
7841 = (19 * 412) + 13 ;
hence not 19 divides 7841 by NAT_4:9; :: thesis:
7841 = (23 * 340) + 21 ;
hence not 23 divides 7841 by NAT_4:9; :: thesis:
7841 = (29 * 270) + 11 ;
hence not 29 divides 7841 by NAT_4:9; :: thesis:
7841 = (31 * 252) + 29 ;
hence not 31 divides 7841 by NAT_4:9; :: thesis:
7841 = (37 * 211) + 34 ;
hence not 37 divides 7841 by NAT_4:9; :: thesis:
7841 = (41 * 191) + 10 ;
hence not 41 divides 7841 by NAT_4:9; :: thesis:
7841 = (43 * 182) + 15 ;
hence not 43 divides 7841 by NAT_4:9; :: thesis:
7841 = (47 * 166) + 39 ;
hence not 47 divides 7841 by NAT_4:9; :: thesis:
7841 = (53 * 147) + 50 ;
hence not 53 divides 7841 by NAT_4:9; :: thesis:
7841 = (59 * 132) + 53 ;
hence not 59 divides 7841 by NAT_4:9; :: thesis:
7841 = (61 * 128) + 33 ;
hence not 61 divides 7841 by NAT_4:9; :: thesis:
7841 = (67 * 117) + 2 ;
hence not 67 divides 7841 by NAT_4:9; :: thesis:
7841 = (71 * 110) + 31 ;
hence not 71 divides 7841 by NAT_4:9; :: thesis:
7841 = (73 * 107) + 30 ;
hence not 73 divides 7841 by NAT_4:9; :: thesis:
7841 = (79 * 99) + 20 ;
hence not 79 divides 7841 by NAT_4:9; :: thesis:
7841 = (83 * 94) + 39 ;
hence not 83 divides 7841 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7841 & n is prime holds
not n divides 7841 by XPRIMET1:46;
hence 7841 is prime by NAT_4:14; :: thesis: