8933 = (2 * 4466) + 1 ;
hence not 2 divides 8933 by NAT_4:9; :: thesis:
8933 = (3 * 2977) + 2 ;
hence not 3 divides 8933 by NAT_4:9; :: thesis:
8933 = (5 * 1786) + 3 ;
hence not 5 divides 8933 by NAT_4:9; :: thesis:
8933 = (7 * 1276) + 1 ;
hence not 7 divides 8933 by NAT_4:9; :: thesis:
8933 = (11 * 812) + 1 ;
hence not 11 divides 8933 by NAT_4:9; :: thesis:
8933 = (13 * 687) + 2 ;
hence not 13 divides 8933 by NAT_4:9; :: thesis:
8933 = (17 * 525) + 8 ;
hence not 17 divides 8933 by NAT_4:9; :: thesis:
8933 = (19 * 470) + 3 ;
hence not 19 divides 8933 by NAT_4:9; :: thesis:
8933 = (23 * 388) + 9 ;
hence not 23 divides 8933 by NAT_4:9; :: thesis:
8933 = (29 * 308) + 1 ;
hence not 29 divides 8933 by NAT_4:9; :: thesis:
8933 = (31 * 288) + 5 ;
hence not 31 divides 8933 by NAT_4:9; :: thesis:
8933 = (37 * 241) + 16 ;
hence not 37 divides 8933 by NAT_4:9; :: thesis:
8933 = (41 * 217) + 36 ;
hence not 41 divides 8933 by NAT_4:9; :: thesis:
8933 = (43 * 207) + 32 ;
hence not 43 divides 8933 by NAT_4:9; :: thesis:
8933 = (47 * 190) + 3 ;
hence not 47 divides 8933 by NAT_4:9; :: thesis:
8933 = (53 * 168) + 29 ;
hence not 53 divides 8933 by NAT_4:9; :: thesis:
8933 = (59 * 151) + 24 ;
hence not 59 divides 8933 by NAT_4:9; :: thesis:
8933 = (61 * 146) + 27 ;
hence not 61 divides 8933 by NAT_4:9; :: thesis:
8933 = (67 * 133) + 22 ;
hence not 67 divides 8933 by NAT_4:9; :: thesis:
8933 = (71 * 125) + 58 ;
hence not 71 divides 8933 by NAT_4:9; :: thesis:
8933 = (73 * 122) + 27 ;
hence not 73 divides 8933 by NAT_4:9; :: thesis:
8933 = (79 * 113) + 6 ;
hence not 79 divides 8933 by NAT_4:9; :: thesis:
8933 = (83 * 107) + 52 ;
hence not 83 divides 8933 by NAT_4:9; :: thesis:
8933 = (89 * 100) + 33 ;
hence not 89 divides 8933 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8933 & n is prime holds
not n divides 8933 by XPRIMET1:48;
hence 8933 is prime by NAT_4:14; :: thesis: