8093 = (2 * 4046) + 1 ;
hence not 2 divides 8093 by NAT_4:9; :: thesis:
8093 = (3 * 2697) + 2 ;
hence not 3 divides 8093 by NAT_4:9; :: thesis:
8093 = (5 * 1618) + 3 ;
hence not 5 divides 8093 by NAT_4:9; :: thesis:
8093 = (7 * 1156) + 1 ;
hence not 7 divides 8093 by NAT_4:9; :: thesis:
8093 = (11 * 735) + 8 ;
hence not 11 divides 8093 by NAT_4:9; :: thesis:
8093 = (13 * 622) + 7 ;
hence not 13 divides 8093 by NAT_4:9; :: thesis:
8093 = (17 * 476) + 1 ;
hence not 17 divides 8093 by NAT_4:9; :: thesis:
8093 = (19 * 425) + 18 ;
hence not 19 divides 8093 by NAT_4:9; :: thesis:
8093 = (23 * 351) + 20 ;
hence not 23 divides 8093 by NAT_4:9; :: thesis:
8093 = (29 * 279) + 2 ;
hence not 29 divides 8093 by NAT_4:9; :: thesis:
8093 = (31 * 261) + 2 ;
hence not 31 divides 8093 by NAT_4:9; :: thesis:
8093 = (37 * 218) + 27 ;
hence not 37 divides 8093 by NAT_4:9; :: thesis:
8093 = (41 * 197) + 16 ;
hence not 41 divides 8093 by NAT_4:9; :: thesis:
8093 = (43 * 188) + 9 ;
hence not 43 divides 8093 by NAT_4:9; :: thesis:
8093 = (47 * 172) + 9 ;
hence not 47 divides 8093 by NAT_4:9; :: thesis:
8093 = (53 * 152) + 37 ;
hence not 53 divides 8093 by NAT_4:9; :: thesis:
8093 = (59 * 137) + 10 ;
hence not 59 divides 8093 by NAT_4:9; :: thesis:
8093 = (61 * 132) + 41 ;
hence not 61 divides 8093 by NAT_4:9; :: thesis:
8093 = (67 * 120) + 53 ;
hence not 67 divides 8093 by NAT_4:9; :: thesis:
8093 = (71 * 113) + 70 ;
hence not 71 divides 8093 by NAT_4:9; :: thesis:
8093 = (73 * 110) + 63 ;
hence not 73 divides 8093 by NAT_4:9; :: thesis:
8093 = (79 * 102) + 35 ;
hence not 79 divides 8093 by NAT_4:9; :: thesis:
8093 = (83 * 97) + 42 ;
hence not 83 divides 8093 by NAT_4:9; :: thesis:
8093 = (89 * 90) + 83 ;
hence not 89 divides 8093 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8093 & n is prime holds
not n divides 8093 by XPRIMET1:48;
hence 8093 is prime by NAT_4:14; :: thesis: