8353 = (2 * 4176) + 1 ;
hence not 2 divides 8353 by NAT_4:9; :: thesis:
8353 = (3 * 2784) + 1 ;
hence not 3 divides 8353 by NAT_4:9; :: thesis:
8353 = (5 * 1670) + 3 ;
hence not 5 divides 8353 by NAT_4:9; :: thesis:
8353 = (7 * 1193) + 2 ;
hence not 7 divides 8353 by NAT_4:9; :: thesis:
8353 = (11 * 759) + 4 ;
hence not 11 divides 8353 by NAT_4:9; :: thesis:
8353 = (13 * 642) + 7 ;
hence not 13 divides 8353 by NAT_4:9; :: thesis:
8353 = (17 * 491) + 6 ;
hence not 17 divides 8353 by NAT_4:9; :: thesis:
8353 = (19 * 439) + 12 ;
hence not 19 divides 8353 by NAT_4:9; :: thesis:
8353 = (23 * 363) + 4 ;
hence not 23 divides 8353 by NAT_4:9; :: thesis:
8353 = (29 * 288) + 1 ;
hence not 29 divides 8353 by NAT_4:9; :: thesis:
8353 = (31 * 269) + 14 ;
hence not 31 divides 8353 by NAT_4:9; :: thesis:
8353 = (37 * 225) + 28 ;
hence not 37 divides 8353 by NAT_4:9; :: thesis:
8353 = (41 * 203) + 30 ;
hence not 41 divides 8353 by NAT_4:9; :: thesis:
8353 = (43 * 194) + 11 ;
hence not 43 divides 8353 by NAT_4:9; :: thesis:
8353 = (47 * 177) + 34 ;
hence not 47 divides 8353 by NAT_4:9; :: thesis:
8353 = (53 * 157) + 32 ;
hence not 53 divides 8353 by NAT_4:9; :: thesis:
8353 = (59 * 141) + 34 ;
hence not 59 divides 8353 by NAT_4:9; :: thesis:
8353 = (61 * 136) + 57 ;
hence not 61 divides 8353 by NAT_4:9; :: thesis:
8353 = (67 * 124) + 45 ;
hence not 67 divides 8353 by NAT_4:9; :: thesis:
8353 = (71 * 117) + 46 ;
hence not 71 divides 8353 by NAT_4:9; :: thesis:
8353 = (73 * 114) + 31 ;
hence not 73 divides 8353 by NAT_4:9; :: thesis:
8353 = (79 * 105) + 58 ;
hence not 79 divides 8353 by NAT_4:9; :: thesis:
8353 = (83 * 100) + 53 ;
hence not 83 divides 8353 by NAT_4:9; :: thesis:
8353 = (89 * 93) + 76 ;
hence not 89 divides 8353 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8353 & n is prime holds
not n divides 8353 by XPRIMET1:48;
hence 8353 is prime by NAT_4:14; :: thesis: