7907 = (2 * 3953) + 1 ;
hence not 2 divides 7907 by NAT_4:9; :: thesis:
7907 = (3 * 2635) + 2 ;
hence not 3 divides 7907 by NAT_4:9; :: thesis:
7907 = (5 * 1581) + 2 ;
hence not 5 divides 7907 by NAT_4:9; :: thesis:
7907 = (7 * 1129) + 4 ;
hence not 7 divides 7907 by NAT_4:9; :: thesis:
7907 = (11 * 718) + 9 ;
hence not 11 divides 7907 by NAT_4:9; :: thesis:
7907 = (13 * 608) + 3 ;
hence not 13 divides 7907 by NAT_4:9; :: thesis:
7907 = (17 * 465) + 2 ;
hence not 17 divides 7907 by NAT_4:9; :: thesis:
7907 = (19 * 416) + 3 ;
hence not 19 divides 7907 by NAT_4:9; :: thesis:
7907 = (23 * 343) + 18 ;
hence not 23 divides 7907 by NAT_4:9; :: thesis:
7907 = (29 * 272) + 19 ;
hence not 29 divides 7907 by NAT_4:9; :: thesis:
7907 = (31 * 255) + 2 ;
hence not 31 divides 7907 by NAT_4:9; :: thesis:
7907 = (37 * 213) + 26 ;
hence not 37 divides 7907 by NAT_4:9; :: thesis:
7907 = (41 * 192) + 35 ;
hence not 41 divides 7907 by NAT_4:9; :: thesis:
7907 = (43 * 183) + 38 ;
hence not 43 divides 7907 by NAT_4:9; :: thesis:
7907 = (47 * 168) + 11 ;
hence not 47 divides 7907 by NAT_4:9; :: thesis:
7907 = (53 * 149) + 10 ;
hence not 53 divides 7907 by NAT_4:9; :: thesis:
7907 = (59 * 134) + 1 ;
hence not 59 divides 7907 by NAT_4:9; :: thesis:
7907 = (61 * 129) + 38 ;
hence not 61 divides 7907 by NAT_4:9; :: thesis:
7907 = (67 * 118) + 1 ;
hence not 67 divides 7907 by NAT_4:9; :: thesis:
7907 = (71 * 111) + 26 ;
hence not 71 divides 7907 by NAT_4:9; :: thesis:
7907 = (73 * 108) + 23 ;
hence not 73 divides 7907 by NAT_4:9; :: thesis:
7907 = (79 * 100) + 7 ;
hence not 79 divides 7907 by NAT_4:9; :: thesis:
7907 = (83 * 95) + 22 ;
hence not 83 divides 7907 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7907 & n is prime holds
not n divides 7907 by XPRIMET1:46;
hence 7907 is prime by NAT_4:14; :: thesis: