6781 = (2 * 3390) + 1 ;
hence not 2 divides 6781 by NAT_4:9; :: thesis:
6781 = (3 * 2260) + 1 ;
hence not 3 divides 6781 by NAT_4:9; :: thesis:
6781 = (5 * 1356) + 1 ;
hence not 5 divides 6781 by NAT_4:9; :: thesis:
6781 = (7 * 968) + 5 ;
hence not 7 divides 6781 by NAT_4:9; :: thesis:
6781 = (11 * 616) + 5 ;
hence not 11 divides 6781 by NAT_4:9; :: thesis:
6781 = (13 * 521) + 8 ;
hence not 13 divides 6781 by NAT_4:9; :: thesis:
6781 = (17 * 398) + 15 ;
hence not 17 divides 6781 by NAT_4:9; :: thesis:
6781 = (19 * 356) + 17 ;
hence not 19 divides 6781 by NAT_4:9; :: thesis:
6781 = (23 * 294) + 19 ;
hence not 23 divides 6781 by NAT_4:9; :: thesis:
6781 = (29 * 233) + 24 ;
hence not 29 divides 6781 by NAT_4:9; :: thesis:
6781 = (31 * 218) + 23 ;
hence not 31 divides 6781 by NAT_4:9; :: thesis:
6781 = (37 * 183) + 10 ;
hence not 37 divides 6781 by NAT_4:9; :: thesis:
6781 = (41 * 165) + 16 ;
hence not 41 divides 6781 by NAT_4:9; :: thesis:
6781 = (43 * 157) + 30 ;
hence not 43 divides 6781 by NAT_4:9; :: thesis:
6781 = (47 * 144) + 13 ;
hence not 47 divides 6781 by NAT_4:9; :: thesis:
6781 = (53 * 127) + 50 ;
hence not 53 divides 6781 by NAT_4:9; :: thesis:
6781 = (59 * 114) + 55 ;
hence not 59 divides 6781 by NAT_4:9; :: thesis:
6781 = (61 * 111) + 10 ;
hence not 61 divides 6781 by NAT_4:9; :: thesis:
6781 = (67 * 101) + 14 ;
hence not 67 divides 6781 by NAT_4:9; :: thesis:
6781 = (71 * 95) + 36 ;
hence not 71 divides 6781 by NAT_4:9; :: thesis:
6781 = (73 * 92) + 65 ;
hence not 73 divides 6781 by NAT_4:9; :: thesis:
6781 = (79 * 85) + 66 ;
hence not 79 divides 6781 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6781 & n is prime holds
not n divides 6781 by XPRIMET1:44;
hence 6781 is prime by NAT_4:14; :: thesis: