8923 = (2 * 4461) + 1 ;
hence not 2 divides 8923 by NAT_4:9; :: thesis:
8923 = (3 * 2974) + 1 ;
hence not 3 divides 8923 by NAT_4:9; :: thesis:
8923 = (5 * 1784) + 3 ;
hence not 5 divides 8923 by NAT_4:9; :: thesis:
8923 = (7 * 1274) + 5 ;
hence not 7 divides 8923 by NAT_4:9; :: thesis:
8923 = (11 * 811) + 2 ;
hence not 11 divides 8923 by NAT_4:9; :: thesis:
8923 = (13 * 686) + 5 ;
hence not 13 divides 8923 by NAT_4:9; :: thesis:
8923 = (17 * 524) + 15 ;
hence not 17 divides 8923 by NAT_4:9; :: thesis:
8923 = (19 * 469) + 12 ;
hence not 19 divides 8923 by NAT_4:9; :: thesis:
8923 = (23 * 387) + 22 ;
hence not 23 divides 8923 by NAT_4:9; :: thesis:
8923 = (29 * 307) + 20 ;
hence not 29 divides 8923 by NAT_4:9; :: thesis:
8923 = (31 * 287) + 26 ;
hence not 31 divides 8923 by NAT_4:9; :: thesis:
8923 = (37 * 241) + 6 ;
hence not 37 divides 8923 by NAT_4:9; :: thesis:
8923 = (41 * 217) + 26 ;
hence not 41 divides 8923 by NAT_4:9; :: thesis:
8923 = (43 * 207) + 22 ;
hence not 43 divides 8923 by NAT_4:9; :: thesis:
8923 = (47 * 189) + 40 ;
hence not 47 divides 8923 by NAT_4:9; :: thesis:
8923 = (53 * 168) + 19 ;
hence not 53 divides 8923 by NAT_4:9; :: thesis:
8923 = (59 * 151) + 14 ;
hence not 59 divides 8923 by NAT_4:9; :: thesis:
8923 = (61 * 146) + 17 ;
hence not 61 divides 8923 by NAT_4:9; :: thesis:
8923 = (67 * 133) + 12 ;
hence not 67 divides 8923 by NAT_4:9; :: thesis:
8923 = (71 * 125) + 48 ;
hence not 71 divides 8923 by NAT_4:9; :: thesis:
8923 = (73 * 122) + 17 ;
hence not 73 divides 8923 by NAT_4:9; :: thesis:
8923 = (79 * 112) + 75 ;
hence not 79 divides 8923 by NAT_4:9; :: thesis:
8923 = (83 * 107) + 42 ;
hence not 83 divides 8923 by NAT_4:9; :: thesis:
8923 = (89 * 100) + 23 ;
hence not 89 divides 8923 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8923 & n is prime holds
not n divides 8923 by XPRIMET1:48;
hence 8923 is prime by NAT_4:14; :: thesis: