8563 = (2 * 4281) + 1 ;
hence not 2 divides 8563 by NAT_4:9; :: thesis:
8563 = (3 * 2854) + 1 ;
hence not 3 divides 8563 by NAT_4:9; :: thesis:
8563 = (5 * 1712) + 3 ;
hence not 5 divides 8563 by NAT_4:9; :: thesis:
8563 = (7 * 1223) + 2 ;
hence not 7 divides 8563 by NAT_4:9; :: thesis:
8563 = (11 * 778) + 5 ;
hence not 11 divides 8563 by NAT_4:9; :: thesis:
8563 = (13 * 658) + 9 ;
hence not 13 divides 8563 by NAT_4:9; :: thesis:
8563 = (17 * 503) + 12 ;
hence not 17 divides 8563 by NAT_4:9; :: thesis:
8563 = (19 * 450) + 13 ;
hence not 19 divides 8563 by NAT_4:9; :: thesis:
8563 = (23 * 372) + 7 ;
hence not 23 divides 8563 by NAT_4:9; :: thesis:
8563 = (29 * 295) + 8 ;
hence not 29 divides 8563 by NAT_4:9; :: thesis:
8563 = (31 * 276) + 7 ;
hence not 31 divides 8563 by NAT_4:9; :: thesis:
8563 = (37 * 231) + 16 ;
hence not 37 divides 8563 by NAT_4:9; :: thesis:
8563 = (41 * 208) + 35 ;
hence not 41 divides 8563 by NAT_4:9; :: thesis:
8563 = (43 * 199) + 6 ;
hence not 43 divides 8563 by NAT_4:9; :: thesis:
8563 = (47 * 182) + 9 ;
hence not 47 divides 8563 by NAT_4:9; :: thesis:
8563 = (53 * 161) + 30 ;
hence not 53 divides 8563 by NAT_4:9; :: thesis:
8563 = (59 * 145) + 8 ;
hence not 59 divides 8563 by NAT_4:9; :: thesis:
8563 = (61 * 140) + 23 ;
hence not 61 divides 8563 by NAT_4:9; :: thesis:
8563 = (67 * 127) + 54 ;
hence not 67 divides 8563 by NAT_4:9; :: thesis:
8563 = (71 * 120) + 43 ;
hence not 71 divides 8563 by NAT_4:9; :: thesis:
8563 = (73 * 117) + 22 ;
hence not 73 divides 8563 by NAT_4:9; :: thesis:
8563 = (79 * 108) + 31 ;
hence not 79 divides 8563 by NAT_4:9; :: thesis:
8563 = (83 * 103) + 14 ;
hence not 83 divides 8563 by NAT_4:9; :: thesis:
8563 = (89 * 96) + 19 ;
hence not 89 divides 8563 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8563 & n is prime holds
not n divides 8563 by XPRIMET1:48;
hence 8563 is prime by NAT_4:14; :: thesis: