9907 = (2 * 4953) + 1 ;
hence not 2 divides 9907 by NAT_4:9; :: thesis:
9907 = (3 * 3302) + 1 ;
hence not 3 divides 9907 by NAT_4:9; :: thesis:
9907 = (5 * 1981) + 2 ;
hence not 5 divides 9907 by NAT_4:9; :: thesis:
9907 = (7 * 1415) + 2 ;
hence not 7 divides 9907 by NAT_4:9; :: thesis:
9907 = (11 * 900) + 7 ;
hence not 11 divides 9907 by NAT_4:9; :: thesis:
9907 = (13 * 762) + 1 ;
hence not 13 divides 9907 by NAT_4:9; :: thesis:
9907 = (17 * 582) + 13 ;
hence not 17 divides 9907 by NAT_4:9; :: thesis:
9907 = (19 * 521) + 8 ;
hence not 19 divides 9907 by NAT_4:9; :: thesis:
9907 = (23 * 430) + 17 ;
hence not 23 divides 9907 by NAT_4:9; :: thesis:
9907 = (29 * 341) + 18 ;
hence not 29 divides 9907 by NAT_4:9; :: thesis:
9907 = (31 * 319) + 18 ;
hence not 31 divides 9907 by NAT_4:9; :: thesis:
9907 = (37 * 267) + 28 ;
hence not 37 divides 9907 by NAT_4:9; :: thesis:
9907 = (41 * 241) + 26 ;
hence not 41 divides 9907 by NAT_4:9; :: thesis:
9907 = (43 * 230) + 17 ;
hence not 43 divides 9907 by NAT_4:9; :: thesis:
9907 = (47 * 210) + 37 ;
hence not 47 divides 9907 by NAT_4:9; :: thesis:
9907 = (53 * 186) + 49 ;
hence not 53 divides 9907 by NAT_4:9; :: thesis:
9907 = (59 * 167) + 54 ;
hence not 59 divides 9907 by NAT_4:9; :: thesis:
9907 = (61 * 162) + 25 ;
hence not 61 divides 9907 by NAT_4:9; :: thesis:
9907 = (67 * 147) + 58 ;
hence not 67 divides 9907 by NAT_4:9; :: thesis:
9907 = (71 * 139) + 38 ;
hence not 71 divides 9907 by NAT_4:9; :: thesis:
9907 = (73 * 135) + 52 ;
hence not 73 divides 9907 by NAT_4:9; :: thesis:
9907 = (79 * 125) + 32 ;
hence not 79 divides 9907 by NAT_4:9; :: thesis:
9907 = (83 * 119) + 30 ;
hence not 83 divides 9907 by NAT_4:9; :: thesis:
9907 = (89 * 111) + 28 ;
hence not 89 divides 9907 by NAT_4:9; :: thesis:
9907 = (97 * 102) + 13 ;
hence not 97 divides 9907 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9907 & n is prime holds
not n divides 9907 by XPRIMET1:50;
hence 9907 is prime by NAT_4:14; :: thesis: