6907 = (2 * 3453) + 1 ;
hence not 2 divides 6907 by NAT_4:9; :: thesis:
6907 = (3 * 2302) + 1 ;
hence not 3 divides 6907 by NAT_4:9; :: thesis:
6907 = (5 * 1381) + 2 ;
hence not 5 divides 6907 by NAT_4:9; :: thesis:
6907 = (7 * 986) + 5 ;
hence not 7 divides 6907 by NAT_4:9; :: thesis:
6907 = (11 * 627) + 10 ;
hence not 11 divides 6907 by NAT_4:9; :: thesis:
6907 = (13 * 531) + 4 ;
hence not 13 divides 6907 by NAT_4:9; :: thesis:
6907 = (17 * 406) + 5 ;
hence not 17 divides 6907 by NAT_4:9; :: thesis:
6907 = (19 * 363) + 10 ;
hence not 19 divides 6907 by NAT_4:9; :: thesis:
6907 = (23 * 300) + 7 ;
hence not 23 divides 6907 by NAT_4:9; :: thesis:
6907 = (29 * 238) + 5 ;
hence not 29 divides 6907 by NAT_4:9; :: thesis:
6907 = (31 * 222) + 25 ;
hence not 31 divides 6907 by NAT_4:9; :: thesis:
6907 = (37 * 186) + 25 ;
hence not 37 divides 6907 by NAT_4:9; :: thesis:
6907 = (41 * 168) + 19 ;
hence not 41 divides 6907 by NAT_4:9; :: thesis:
6907 = (43 * 160) + 27 ;
hence not 43 divides 6907 by NAT_4:9; :: thesis:
6907 = (47 * 146) + 45 ;
hence not 47 divides 6907 by NAT_4:9; :: thesis:
6907 = (53 * 130) + 17 ;
hence not 53 divides 6907 by NAT_4:9; :: thesis:
6907 = (59 * 117) + 4 ;
hence not 59 divides 6907 by NAT_4:9; :: thesis:
6907 = (61 * 113) + 14 ;
hence not 61 divides 6907 by NAT_4:9; :: thesis:
6907 = (67 * 103) + 6 ;
hence not 67 divides 6907 by NAT_4:9; :: thesis:
6907 = (71 * 97) + 20 ;
hence not 71 divides 6907 by NAT_4:9; :: thesis:
6907 = (73 * 94) + 45 ;
hence not 73 divides 6907 by NAT_4:9; :: thesis:
6907 = (79 * 87) + 34 ;
hence not 79 divides 6907 by NAT_4:9; :: thesis:
6907 = (83 * 83) + 18 ;
hence not 83 divides 6907 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6907 & n is prime holds
not n divides 6907 by XPRIMET1:46;
hence 6907 is prime by NAT_4:14; :: thesis: