4817 = (2 * 2408) + 1 ;
hence not 2 divides 4817 by NAT_4:9; :: thesis:
4817 = (3 * 1605) + 2 ;
hence not 3 divides 4817 by NAT_4:9; :: thesis:
4817 = (5 * 963) + 2 ;
hence not 5 divides 4817 by NAT_4:9; :: thesis:
4817 = (7 * 688) + 1 ;
hence not 7 divides 4817 by NAT_4:9; :: thesis:
4817 = (11 * 437) + 10 ;
hence not 11 divides 4817 by NAT_4:9; :: thesis:
4817 = (13 * 370) + 7 ;
hence not 13 divides 4817 by NAT_4:9; :: thesis:
4817 = (17 * 283) + 6 ;
hence not 17 divides 4817 by NAT_4:9; :: thesis:
4817 = (19 * 253) + 10 ;
hence not 19 divides 4817 by NAT_4:9; :: thesis:
4817 = (23 * 209) + 10 ;
hence not 23 divides 4817 by NAT_4:9; :: thesis:
4817 = (29 * 166) + 3 ;
hence not 29 divides 4817 by NAT_4:9; :: thesis:
4817 = (31 * 155) + 12 ;
hence not 31 divides 4817 by NAT_4:9; :: thesis:
4817 = (37 * 130) + 7 ;
hence not 37 divides 4817 by NAT_4:9; :: thesis:
4817 = (41 * 117) + 20 ;
hence not 41 divides 4817 by NAT_4:9; :: thesis:
4817 = (43 * 112) + 1 ;
hence not 43 divides 4817 by NAT_4:9; :: thesis:
4817 = (47 * 102) + 23 ;
hence not 47 divides 4817 by NAT_4:9; :: thesis:
4817 = (53 * 90) + 47 ;
hence not 53 divides 4817 by NAT_4:9; :: thesis:
4817 = (59 * 81) + 38 ;
hence not 59 divides 4817 by NAT_4:9; :: thesis:
4817 = (61 * 78) + 59 ;
hence not 61 divides 4817 by NAT_4:9; :: thesis:
4817 = (67 * 71) + 60 ;
hence not 67 divides 4817 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4817 & n is prime holds
not n divides 4817 by XPRIMET1:38;
hence 4817 is prime by NAT_4:14; :: thesis: