6719 = (2 * 3359) + 1 ;
hence not 2 divides 6719 by NAT_4:9; :: thesis:
6719 = (3 * 2239) + 2 ;
hence not 3 divides 6719 by NAT_4:9; :: thesis:
6719 = (5 * 1343) + 4 ;
hence not 5 divides 6719 by NAT_4:9; :: thesis:
6719 = (7 * 959) + 6 ;
hence not 7 divides 6719 by NAT_4:9; :: thesis:
6719 = (11 * 610) + 9 ;
hence not 11 divides 6719 by NAT_4:9; :: thesis:
6719 = (13 * 516) + 11 ;
hence not 13 divides 6719 by NAT_4:9; :: thesis:
6719 = (17 * 395) + 4 ;
hence not 17 divides 6719 by NAT_4:9; :: thesis:
6719 = (19 * 353) + 12 ;
hence not 19 divides 6719 by NAT_4:9; :: thesis:
6719 = (23 * 292) + 3 ;
hence not 23 divides 6719 by NAT_4:9; :: thesis:
6719 = (29 * 231) + 20 ;
hence not 29 divides 6719 by NAT_4:9; :: thesis:
6719 = (31 * 216) + 23 ;
hence not 31 divides 6719 by NAT_4:9; :: thesis:
6719 = (37 * 181) + 22 ;
hence not 37 divides 6719 by NAT_4:9; :: thesis:
6719 = (41 * 163) + 36 ;
hence not 41 divides 6719 by NAT_4:9; :: thesis:
6719 = (43 * 156) + 11 ;
hence not 43 divides 6719 by NAT_4:9; :: thesis:
6719 = (47 * 142) + 45 ;
hence not 47 divides 6719 by NAT_4:9; :: thesis:
6719 = (53 * 126) + 41 ;
hence not 53 divides 6719 by NAT_4:9; :: thesis:
6719 = (59 * 113) + 52 ;
hence not 59 divides 6719 by NAT_4:9; :: thesis:
6719 = (61 * 110) + 9 ;
hence not 61 divides 6719 by NAT_4:9; :: thesis:
6719 = (67 * 100) + 19 ;
hence not 67 divides 6719 by NAT_4:9; :: thesis:
6719 = (71 * 94) + 45 ;
hence not 71 divides 6719 by NAT_4:9; :: thesis:
6719 = (73 * 92) + 3 ;
hence not 73 divides 6719 by NAT_4:9; :: thesis:
6719 = (79 * 85) + 4 ;
hence not 79 divides 6719 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6719 & n is prime holds
not n divides 6719 by XPRIMET1:44;
hence 6719 is prime by NAT_4:14; :: thesis: