7109 = (2 * 3554) + 1 ;
hence not 2 divides 7109 by NAT_4:9; :: thesis:
7109 = (3 * 2369) + 2 ;
hence not 3 divides 7109 by NAT_4:9; :: thesis:
7109 = (5 * 1421) + 4 ;
hence not 5 divides 7109 by NAT_4:9; :: thesis:
7109 = (7 * 1015) + 4 ;
hence not 7 divides 7109 by NAT_4:9; :: thesis:
7109 = (11 * 646) + 3 ;
hence not 11 divides 7109 by NAT_4:9; :: thesis:
7109 = (13 * 546) + 11 ;
hence not 13 divides 7109 by NAT_4:9; :: thesis:
7109 = (17 * 418) + 3 ;
hence not 17 divides 7109 by NAT_4:9; :: thesis:
7109 = (19 * 374) + 3 ;
hence not 19 divides 7109 by NAT_4:9; :: thesis:
7109 = (23 * 309) + 2 ;
hence not 23 divides 7109 by NAT_4:9; :: thesis:
7109 = (29 * 245) + 4 ;
hence not 29 divides 7109 by NAT_4:9; :: thesis:
7109 = (31 * 229) + 10 ;
hence not 31 divides 7109 by NAT_4:9; :: thesis:
7109 = (37 * 192) + 5 ;
hence not 37 divides 7109 by NAT_4:9; :: thesis:
7109 = (41 * 173) + 16 ;
hence not 41 divides 7109 by NAT_4:9; :: thesis:
7109 = (43 * 165) + 14 ;
hence not 43 divides 7109 by NAT_4:9; :: thesis:
7109 = (47 * 151) + 12 ;
hence not 47 divides 7109 by NAT_4:9; :: thesis:
7109 = (53 * 134) + 7 ;
hence not 53 divides 7109 by NAT_4:9; :: thesis:
7109 = (59 * 120) + 29 ;
hence not 59 divides 7109 by NAT_4:9; :: thesis:
7109 = (61 * 116) + 33 ;
hence not 61 divides 7109 by NAT_4:9; :: thesis:
7109 = (67 * 106) + 7 ;
hence not 67 divides 7109 by NAT_4:9; :: thesis:
7109 = (71 * 100) + 9 ;
hence not 71 divides 7109 by NAT_4:9; :: thesis:
7109 = (73 * 97) + 28 ;
hence not 73 divides 7109 by NAT_4:9; :: thesis:
7109 = (79 * 89) + 78 ;
hence not 79 divides 7109 by NAT_4:9; :: thesis:
7109 = (83 * 85) + 54 ;
hence not 83 divides 7109 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7109 & n is prime holds
not n divides 7109 by XPRIMET1:46;
hence 7109 is prime by NAT_4:14; :: thesis: