8167 = (2 * 4083) + 1 ;
hence not 2 divides 8167 by NAT_4:9; :: thesis:
8167 = (3 * 2722) + 1 ;
hence not 3 divides 8167 by NAT_4:9; :: thesis:
8167 = (5 * 1633) + 2 ;
hence not 5 divides 8167 by NAT_4:9; :: thesis:
8167 = (7 * 1166) + 5 ;
hence not 7 divides 8167 by NAT_4:9; :: thesis:
8167 = (11 * 742) + 5 ;
hence not 11 divides 8167 by NAT_4:9; :: thesis:
8167 = (13 * 628) + 3 ;
hence not 13 divides 8167 by NAT_4:9; :: thesis:
8167 = (17 * 480) + 7 ;
hence not 17 divides 8167 by NAT_4:9; :: thesis:
8167 = (19 * 429) + 16 ;
hence not 19 divides 8167 by NAT_4:9; :: thesis:
8167 = (23 * 355) + 2 ;
hence not 23 divides 8167 by NAT_4:9; :: thesis:
8167 = (29 * 281) + 18 ;
hence not 29 divides 8167 by NAT_4:9; :: thesis:
8167 = (31 * 263) + 14 ;
hence not 31 divides 8167 by NAT_4:9; :: thesis:
8167 = (37 * 220) + 27 ;
hence not 37 divides 8167 by NAT_4:9; :: thesis:
8167 = (41 * 199) + 8 ;
hence not 41 divides 8167 by NAT_4:9; :: thesis:
8167 = (43 * 189) + 40 ;
hence not 43 divides 8167 by NAT_4:9; :: thesis:
8167 = (47 * 173) + 36 ;
hence not 47 divides 8167 by NAT_4:9; :: thesis:
8167 = (53 * 154) + 5 ;
hence not 53 divides 8167 by NAT_4:9; :: thesis:
8167 = (59 * 138) + 25 ;
hence not 59 divides 8167 by NAT_4:9; :: thesis:
8167 = (61 * 133) + 54 ;
hence not 61 divides 8167 by NAT_4:9; :: thesis:
8167 = (67 * 121) + 60 ;
hence not 67 divides 8167 by NAT_4:9; :: thesis:
8167 = (71 * 115) + 2 ;
hence not 71 divides 8167 by NAT_4:9; :: thesis:
8167 = (73 * 111) + 64 ;
hence not 73 divides 8167 by NAT_4:9; :: thesis:
8167 = (79 * 103) + 30 ;
hence not 79 divides 8167 by NAT_4:9; :: thesis:
8167 = (83 * 98) + 33 ;
hence not 83 divides 8167 by NAT_4:9; :: thesis:
8167 = (89 * 91) + 68 ;
hence not 89 divides 8167 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8167 & n is prime holds
not n divides 8167 by XPRIMET1:48;
hence 8167 is prime by NAT_4:14; :: thesis: