6803 = (2 * 3401) + 1 ;
hence not 2 divides 6803 by NAT_4:9; :: thesis:
6803 = (3 * 2267) + 2 ;
hence not 3 divides 6803 by NAT_4:9; :: thesis:
6803 = (5 * 1360) + 3 ;
hence not 5 divides 6803 by NAT_4:9; :: thesis:
6803 = (7 * 971) + 6 ;
hence not 7 divides 6803 by NAT_4:9; :: thesis:
6803 = (11 * 618) + 5 ;
hence not 11 divides 6803 by NAT_4:9; :: thesis:
6803 = (13 * 523) + 4 ;
hence not 13 divides 6803 by NAT_4:9; :: thesis:
6803 = (17 * 400) + 3 ;
hence not 17 divides 6803 by NAT_4:9; :: thesis:
6803 = (19 * 358) + 1 ;
hence not 19 divides 6803 by NAT_4:9; :: thesis:
6803 = (23 * 295) + 18 ;
hence not 23 divides 6803 by NAT_4:9; :: thesis:
6803 = (29 * 234) + 17 ;
hence not 29 divides 6803 by NAT_4:9; :: thesis:
6803 = (31 * 219) + 14 ;
hence not 31 divides 6803 by NAT_4:9; :: thesis:
6803 = (37 * 183) + 32 ;
hence not 37 divides 6803 by NAT_4:9; :: thesis:
6803 = (41 * 165) + 38 ;
hence not 41 divides 6803 by NAT_4:9; :: thesis:
6803 = (43 * 158) + 9 ;
hence not 43 divides 6803 by NAT_4:9; :: thesis:
6803 = (47 * 144) + 35 ;
hence not 47 divides 6803 by NAT_4:9; :: thesis:
6803 = (53 * 128) + 19 ;
hence not 53 divides 6803 by NAT_4:9; :: thesis:
6803 = (59 * 115) + 18 ;
hence not 59 divides 6803 by NAT_4:9; :: thesis:
6803 = (61 * 111) + 32 ;
hence not 61 divides 6803 by NAT_4:9; :: thesis:
6803 = (67 * 101) + 36 ;
hence not 67 divides 6803 by NAT_4:9; :: thesis:
6803 = (71 * 95) + 58 ;
hence not 71 divides 6803 by NAT_4:9; :: thesis:
6803 = (73 * 93) + 14 ;
hence not 73 divides 6803 by NAT_4:9; :: thesis:
6803 = (79 * 86) + 9 ;
hence not 79 divides 6803 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6803 & n is prime holds
not n divides 6803 by XPRIMET1:44;
hence 6803 is prime by NAT_4:14; :: thesis: