6619 = (2 * 3309) + 1 ;
hence not 2 divides 6619 by NAT_4:9; :: thesis:
6619 = (3 * 2206) + 1 ;
hence not 3 divides 6619 by NAT_4:9; :: thesis:
6619 = (5 * 1323) + 4 ;
hence not 5 divides 6619 by NAT_4:9; :: thesis:
6619 = (7 * 945) + 4 ;
hence not 7 divides 6619 by NAT_4:9; :: thesis:
6619 = (11 * 601) + 8 ;
hence not 11 divides 6619 by NAT_4:9; :: thesis:
6619 = (13 * 509) + 2 ;
hence not 13 divides 6619 by NAT_4:9; :: thesis:
6619 = (17 * 389) + 6 ;
hence not 17 divides 6619 by NAT_4:9; :: thesis:
6619 = (19 * 348) + 7 ;
hence not 19 divides 6619 by NAT_4:9; :: thesis:
6619 = (23 * 287) + 18 ;
hence not 23 divides 6619 by NAT_4:9; :: thesis:
6619 = (29 * 228) + 7 ;
hence not 29 divides 6619 by NAT_4:9; :: thesis:
6619 = (31 * 213) + 16 ;
hence not 31 divides 6619 by NAT_4:9; :: thesis:
6619 = (37 * 178) + 33 ;
hence not 37 divides 6619 by NAT_4:9; :: thesis:
6619 = (41 * 161) + 18 ;
hence not 41 divides 6619 by NAT_4:9; :: thesis:
6619 = (43 * 153) + 40 ;
hence not 43 divides 6619 by NAT_4:9; :: thesis:
6619 = (47 * 140) + 39 ;
hence not 47 divides 6619 by NAT_4:9; :: thesis:
6619 = (53 * 124) + 47 ;
hence not 53 divides 6619 by NAT_4:9; :: thesis:
6619 = (59 * 112) + 11 ;
hence not 59 divides 6619 by NAT_4:9; :: thesis:
6619 = (61 * 108) + 31 ;
hence not 61 divides 6619 by NAT_4:9; :: thesis:
6619 = (67 * 98) + 53 ;
hence not 67 divides 6619 by NAT_4:9; :: thesis:
6619 = (71 * 93) + 16 ;
hence not 71 divides 6619 by NAT_4:9; :: thesis:
6619 = (73 * 90) + 49 ;
hence not 73 divides 6619 by NAT_4:9; :: thesis:
6619 = (79 * 83) + 62 ;
hence not 79 divides 6619 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6619 & n is prime holds
not n divides 6619 by XPRIMET1:44;
hence 6619 is prime by NAT_4:14; :: thesis: