8191 = (2 * 4095) + 1 ;
hence not 2 divides 8191 by NAT_4:9; :: thesis:
8191 = (3 * 2730) + 1 ;
hence not 3 divides 8191 by NAT_4:9; :: thesis:
8191 = (5 * 1638) + 1 ;
hence not 5 divides 8191 by NAT_4:9; :: thesis:
8191 = (7 * 1170) + 1 ;
hence not 7 divides 8191 by NAT_4:9; :: thesis:
8191 = (11 * 744) + 7 ;
hence not 11 divides 8191 by NAT_4:9; :: thesis:
8191 = (13 * 630) + 1 ;
hence not 13 divides 8191 by NAT_4:9; :: thesis:
8191 = (17 * 481) + 14 ;
hence not 17 divides 8191 by NAT_4:9; :: thesis:
8191 = (19 * 431) + 2 ;
hence not 19 divides 8191 by NAT_4:9; :: thesis:
8191 = (23 * 356) + 3 ;
hence not 23 divides 8191 by NAT_4:9; :: thesis:
8191 = (29 * 282) + 13 ;
hence not 29 divides 8191 by NAT_4:9; :: thesis:
8191 = (31 * 264) + 7 ;
hence not 31 divides 8191 by NAT_4:9; :: thesis:
8191 = (37 * 221) + 14 ;
hence not 37 divides 8191 by NAT_4:9; :: thesis:
8191 = (41 * 199) + 32 ;
hence not 41 divides 8191 by NAT_4:9; :: thesis:
8191 = (43 * 190) + 21 ;
hence not 43 divides 8191 by NAT_4:9; :: thesis:
8191 = (47 * 174) + 13 ;
hence not 47 divides 8191 by NAT_4:9; :: thesis:
8191 = (53 * 154) + 29 ;
hence not 53 divides 8191 by NAT_4:9; :: thesis:
8191 = (59 * 138) + 49 ;
hence not 59 divides 8191 by NAT_4:9; :: thesis:
8191 = (61 * 134) + 17 ;
hence not 61 divides 8191 by NAT_4:9; :: thesis:
8191 = (67 * 122) + 17 ;
hence not 67 divides 8191 by NAT_4:9; :: thesis:
8191 = (71 * 115) + 26 ;
hence not 71 divides 8191 by NAT_4:9; :: thesis:
8191 = (73 * 112) + 15 ;
hence not 73 divides 8191 by NAT_4:9; :: thesis:
8191 = (79 * 103) + 54 ;
hence not 79 divides 8191 by NAT_4:9; :: thesis:
8191 = (83 * 98) + 57 ;
hence not 83 divides 8191 by NAT_4:9; :: thesis:
8191 = (89 * 92) + 3 ;
hence not 89 divides 8191 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8191 & n is prime holds
not n divides 8191 by XPRIMET1:48;
hence 8191 is prime by NAT_4:14; :: thesis: