7669 = (2 * 3834) + 1 ;
hence not 2 divides 7669 by NAT_4:9; :: thesis:
7669 = (3 * 2556) + 1 ;
hence not 3 divides 7669 by NAT_4:9; :: thesis:
7669 = (5 * 1533) + 4 ;
hence not 5 divides 7669 by NAT_4:9; :: thesis:
7669 = (7 * 1095) + 4 ;
hence not 7 divides 7669 by NAT_4:9; :: thesis:
7669 = (11 * 697) + 2 ;
hence not 11 divides 7669 by NAT_4:9; :: thesis:
7669 = (13 * 589) + 12 ;
hence not 13 divides 7669 by NAT_4:9; :: thesis:
7669 = (17 * 451) + 2 ;
hence not 17 divides 7669 by NAT_4:9; :: thesis:
7669 = (19 * 403) + 12 ;
hence not 19 divides 7669 by NAT_4:9; :: thesis:
7669 = (23 * 333) + 10 ;
hence not 23 divides 7669 by NAT_4:9; :: thesis:
7669 = (29 * 264) + 13 ;
hence not 29 divides 7669 by NAT_4:9; :: thesis:
7669 = (31 * 247) + 12 ;
hence not 31 divides 7669 by NAT_4:9; :: thesis:
7669 = (37 * 207) + 10 ;
hence not 37 divides 7669 by NAT_4:9; :: thesis:
7669 = (41 * 187) + 2 ;
hence not 41 divides 7669 by NAT_4:9; :: thesis:
7669 = (43 * 178) + 15 ;
hence not 43 divides 7669 by NAT_4:9; :: thesis:
7669 = (47 * 163) + 8 ;
hence not 47 divides 7669 by NAT_4:9; :: thesis:
7669 = (53 * 144) + 37 ;
hence not 53 divides 7669 by NAT_4:9; :: thesis:
7669 = (59 * 129) + 58 ;
hence not 59 divides 7669 by NAT_4:9; :: thesis:
7669 = (61 * 125) + 44 ;
hence not 61 divides 7669 by NAT_4:9; :: thesis:
7669 = (67 * 114) + 31 ;
hence not 67 divides 7669 by NAT_4:9; :: thesis:
7669 = (71 * 108) + 1 ;
hence not 71 divides 7669 by NAT_4:9; :: thesis:
7669 = (73 * 105) + 4 ;
hence not 73 divides 7669 by NAT_4:9; :: thesis:
7669 = (79 * 97) + 6 ;
hence not 79 divides 7669 by NAT_4:9; :: thesis:
7669 = (83 * 92) + 33 ;
hence not 83 divides 7669 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7669 & n is prime holds
not n divides 7669 by XPRIMET1:46;
hence 7669 is prime by NAT_4:14; :: thesis: