6823 = (2 * 3411) + 1 ;
hence not 2 divides 6823 by NAT_4:9; :: thesis:
6823 = (3 * 2274) + 1 ;
hence not 3 divides 6823 by NAT_4:9; :: thesis:
6823 = (5 * 1364) + 3 ;
hence not 5 divides 6823 by NAT_4:9; :: thesis:
6823 = (7 * 974) + 5 ;
hence not 7 divides 6823 by NAT_4:9; :: thesis:
6823 = (11 * 620) + 3 ;
hence not 11 divides 6823 by NAT_4:9; :: thesis:
6823 = (13 * 524) + 11 ;
hence not 13 divides 6823 by NAT_4:9; :: thesis:
6823 = (17 * 401) + 6 ;
hence not 17 divides 6823 by NAT_4:9; :: thesis:
6823 = (19 * 359) + 2 ;
hence not 19 divides 6823 by NAT_4:9; :: thesis:
6823 = (23 * 296) + 15 ;
hence not 23 divides 6823 by NAT_4:9; :: thesis:
6823 = (29 * 235) + 8 ;
hence not 29 divides 6823 by NAT_4:9; :: thesis:
6823 = (31 * 220) + 3 ;
hence not 31 divides 6823 by NAT_4:9; :: thesis:
6823 = (37 * 184) + 15 ;
hence not 37 divides 6823 by NAT_4:9; :: thesis:
6823 = (41 * 166) + 17 ;
hence not 41 divides 6823 by NAT_4:9; :: thesis:
6823 = (43 * 158) + 29 ;
hence not 43 divides 6823 by NAT_4:9; :: thesis:
6823 = (47 * 145) + 8 ;
hence not 47 divides 6823 by NAT_4:9; :: thesis:
6823 = (53 * 128) + 39 ;
hence not 53 divides 6823 by NAT_4:9; :: thesis:
6823 = (59 * 115) + 38 ;
hence not 59 divides 6823 by NAT_4:9; :: thesis:
6823 = (61 * 111) + 52 ;
hence not 61 divides 6823 by NAT_4:9; :: thesis:
6823 = (67 * 101) + 56 ;
hence not 67 divides 6823 by NAT_4:9; :: thesis:
6823 = (71 * 96) + 7 ;
hence not 71 divides 6823 by NAT_4:9; :: thesis:
6823 = (73 * 93) + 34 ;
hence not 73 divides 6823 by NAT_4:9; :: thesis:
6823 = (79 * 86) + 29 ;
hence not 79 divides 6823 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6823 & n is prime holds
not n divides 6823 by XPRIMET1:44;
hence 6823 is prime by NAT_4:14; :: thesis: