8219 = (2 * 4109) + 1 ;
hence not 2 divides 8219 by NAT_4:9; :: thesis:
8219 = (3 * 2739) + 2 ;
hence not 3 divides 8219 by NAT_4:9; :: thesis:
8219 = (5 * 1643) + 4 ;
hence not 5 divides 8219 by NAT_4:9; :: thesis:
8219 = (7 * 1174) + 1 ;
hence not 7 divides 8219 by NAT_4:9; :: thesis:
8219 = (11 * 747) + 2 ;
hence not 11 divides 8219 by NAT_4:9; :: thesis:
8219 = (13 * 632) + 3 ;
hence not 13 divides 8219 by NAT_4:9; :: thesis:
8219 = (17 * 483) + 8 ;
hence not 17 divides 8219 by NAT_4:9; :: thesis:
8219 = (19 * 432) + 11 ;
hence not 19 divides 8219 by NAT_4:9; :: thesis:
8219 = (23 * 357) + 8 ;
hence not 23 divides 8219 by NAT_4:9; :: thesis:
8219 = (29 * 283) + 12 ;
hence not 29 divides 8219 by NAT_4:9; :: thesis:
8219 = (31 * 265) + 4 ;
hence not 31 divides 8219 by NAT_4:9; :: thesis:
8219 = (37 * 222) + 5 ;
hence not 37 divides 8219 by NAT_4:9; :: thesis:
8219 = (41 * 200) + 19 ;
hence not 41 divides 8219 by NAT_4:9; :: thesis:
8219 = (43 * 191) + 6 ;
hence not 43 divides 8219 by NAT_4:9; :: thesis:
8219 = (47 * 174) + 41 ;
hence not 47 divides 8219 by NAT_4:9; :: thesis:
8219 = (53 * 155) + 4 ;
hence not 53 divides 8219 by NAT_4:9; :: thesis:
8219 = (59 * 139) + 18 ;
hence not 59 divides 8219 by NAT_4:9; :: thesis:
8219 = (61 * 134) + 45 ;
hence not 61 divides 8219 by NAT_4:9; :: thesis:
8219 = (67 * 122) + 45 ;
hence not 67 divides 8219 by NAT_4:9; :: thesis:
8219 = (71 * 115) + 54 ;
hence not 71 divides 8219 by NAT_4:9; :: thesis:
8219 = (73 * 112) + 43 ;
hence not 73 divides 8219 by NAT_4:9; :: thesis:
8219 = (79 * 104) + 3 ;
hence not 79 divides 8219 by NAT_4:9; :: thesis:
8219 = (83 * 99) + 2 ;
hence not 83 divides 8219 by NAT_4:9; :: thesis:
8219 = (89 * 92) + 31 ;
hence not 89 divides 8219 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8219 & n is prime holds
not n divides 8219 by XPRIMET1:48;
hence 8219 is prime by NAT_4:14; :: thesis: