8501 = (2 * 4250) + 1 ;
hence not 2 divides 8501 by NAT_4:9; :: thesis:
8501 = (3 * 2833) + 2 ;
hence not 3 divides 8501 by NAT_4:9; :: thesis:
8501 = (5 * 1700) + 1 ;
hence not 5 divides 8501 by NAT_4:9; :: thesis:
8501 = (7 * 1214) + 3 ;
hence not 7 divides 8501 by NAT_4:9; :: thesis:
8501 = (11 * 772) + 9 ;
hence not 11 divides 8501 by NAT_4:9; :: thesis:
8501 = (13 * 653) + 12 ;
hence not 13 divides 8501 by NAT_4:9; :: thesis:
8501 = (17 * 500) + 1 ;
hence not 17 divides 8501 by NAT_4:9; :: thesis:
8501 = (19 * 447) + 8 ;
hence not 19 divides 8501 by NAT_4:9; :: thesis:
8501 = (23 * 369) + 14 ;
hence not 23 divides 8501 by NAT_4:9; :: thesis:
8501 = (29 * 293) + 4 ;
hence not 29 divides 8501 by NAT_4:9; :: thesis:
8501 = (31 * 274) + 7 ;
hence not 31 divides 8501 by NAT_4:9; :: thesis:
8501 = (37 * 229) + 28 ;
hence not 37 divides 8501 by NAT_4:9; :: thesis:
8501 = (41 * 207) + 14 ;
hence not 41 divides 8501 by NAT_4:9; :: thesis:
8501 = (43 * 197) + 30 ;
hence not 43 divides 8501 by NAT_4:9; :: thesis:
8501 = (47 * 180) + 41 ;
hence not 47 divides 8501 by NAT_4:9; :: thesis:
8501 = (53 * 160) + 21 ;
hence not 53 divides 8501 by NAT_4:9; :: thesis:
8501 = (59 * 144) + 5 ;
hence not 59 divides 8501 by NAT_4:9; :: thesis:
8501 = (61 * 139) + 22 ;
hence not 61 divides 8501 by NAT_4:9; :: thesis:
8501 = (67 * 126) + 59 ;
hence not 67 divides 8501 by NAT_4:9; :: thesis:
8501 = (71 * 119) + 52 ;
hence not 71 divides 8501 by NAT_4:9; :: thesis:
8501 = (73 * 116) + 33 ;
hence not 73 divides 8501 by NAT_4:9; :: thesis:
8501 = (79 * 107) + 48 ;
hence not 79 divides 8501 by NAT_4:9; :: thesis:
8501 = (83 * 102) + 35 ;
hence not 83 divides 8501 by NAT_4:9; :: thesis:
8501 = (89 * 95) + 46 ;
hence not 89 divides 8501 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8501 & n is prime holds
not n divides 8501 by XPRIMET1:48;
hence 8501 is prime by NAT_4:14; :: thesis: