9817 = (2 * 4908) + 1 ;
hence not 2 divides 9817 by NAT_4:9; :: thesis:
9817 = (3 * 3272) + 1 ;
hence not 3 divides 9817 by NAT_4:9; :: thesis:
9817 = (5 * 1963) + 2 ;
hence not 5 divides 9817 by NAT_4:9; :: thesis:
9817 = (7 * 1402) + 3 ;
hence not 7 divides 9817 by NAT_4:9; :: thesis:
9817 = (11 * 892) + 5 ;
hence not 11 divides 9817 by NAT_4:9; :: thesis:
9817 = (13 * 755) + 2 ;
hence not 13 divides 9817 by NAT_4:9; :: thesis:
9817 = (17 * 577) + 8 ;
hence not 17 divides 9817 by NAT_4:9; :: thesis:
9817 = (19 * 516) + 13 ;
hence not 19 divides 9817 by NAT_4:9; :: thesis:
9817 = (23 * 426) + 19 ;
hence not 23 divides 9817 by NAT_4:9; :: thesis:
9817 = (29 * 338) + 15 ;
hence not 29 divides 9817 by NAT_4:9; :: thesis:
9817 = (31 * 316) + 21 ;
hence not 31 divides 9817 by NAT_4:9; :: thesis:
9817 = (37 * 265) + 12 ;
hence not 37 divides 9817 by NAT_4:9; :: thesis:
9817 = (41 * 239) + 18 ;
hence not 41 divides 9817 by NAT_4:9; :: thesis:
9817 = (43 * 228) + 13 ;
hence not 43 divides 9817 by NAT_4:9; :: thesis:
9817 = (47 * 208) + 41 ;
hence not 47 divides 9817 by NAT_4:9; :: thesis:
9817 = (53 * 185) + 12 ;
hence not 53 divides 9817 by NAT_4:9; :: thesis:
9817 = (59 * 166) + 23 ;
hence not 59 divides 9817 by NAT_4:9; :: thesis:
9817 = (61 * 160) + 57 ;
hence not 61 divides 9817 by NAT_4:9; :: thesis:
9817 = (67 * 146) + 35 ;
hence not 67 divides 9817 by NAT_4:9; :: thesis:
9817 = (71 * 138) + 19 ;
hence not 71 divides 9817 by NAT_4:9; :: thesis:
9817 = (73 * 134) + 35 ;
hence not 73 divides 9817 by NAT_4:9; :: thesis:
9817 = (79 * 124) + 21 ;
hence not 79 divides 9817 by NAT_4:9; :: thesis:
9817 = (83 * 118) + 23 ;
hence not 83 divides 9817 by NAT_4:9; :: thesis:
9817 = (89 * 110) + 27 ;
hence not 89 divides 9817 by NAT_4:9; :: thesis:
9817 = (97 * 101) + 20 ;
hence not 97 divides 9817 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9817 & n is prime holds
not n divides 9817 by XPRIMET1:50;
hence 9817 is prime by NAT_4:14; :: thesis: