9649 = (2 * 4824) + 1 ;
hence not 2 divides 9649 by NAT_4:9; :: thesis:
9649 = (3 * 3216) + 1 ;
hence not 3 divides 9649 by NAT_4:9; :: thesis:
9649 = (5 * 1929) + 4 ;
hence not 5 divides 9649 by NAT_4:9; :: thesis:
9649 = (7 * 1378) + 3 ;
hence not 7 divides 9649 by NAT_4:9; :: thesis:
9649 = (11 * 877) + 2 ;
hence not 11 divides 9649 by NAT_4:9; :: thesis:
9649 = (13 * 742) + 3 ;
hence not 13 divides 9649 by NAT_4:9; :: thesis:
9649 = (17 * 567) + 10 ;
hence not 17 divides 9649 by NAT_4:9; :: thesis:
9649 = (19 * 507) + 16 ;
hence not 19 divides 9649 by NAT_4:9; :: thesis:
9649 = (23 * 419) + 12 ;
hence not 23 divides 9649 by NAT_4:9; :: thesis:
9649 = (29 * 332) + 21 ;
hence not 29 divides 9649 by NAT_4:9; :: thesis:
9649 = (31 * 311) + 8 ;
hence not 31 divides 9649 by NAT_4:9; :: thesis:
9649 = (37 * 260) + 29 ;
hence not 37 divides 9649 by NAT_4:9; :: thesis:
9649 = (41 * 235) + 14 ;
hence not 41 divides 9649 by NAT_4:9; :: thesis:
9649 = (43 * 224) + 17 ;
hence not 43 divides 9649 by NAT_4:9; :: thesis:
9649 = (47 * 205) + 14 ;
hence not 47 divides 9649 by NAT_4:9; :: thesis:
9649 = (53 * 182) + 3 ;
hence not 53 divides 9649 by NAT_4:9; :: thesis:
9649 = (59 * 163) + 32 ;
hence not 59 divides 9649 by NAT_4:9; :: thesis:
9649 = (61 * 158) + 11 ;
hence not 61 divides 9649 by NAT_4:9; :: thesis:
9649 = (67 * 144) + 1 ;
hence not 67 divides 9649 by NAT_4:9; :: thesis:
9649 = (71 * 135) + 64 ;
hence not 71 divides 9649 by NAT_4:9; :: thesis:
9649 = (73 * 132) + 13 ;
hence not 73 divides 9649 by NAT_4:9; :: thesis:
9649 = (79 * 122) + 11 ;
hence not 79 divides 9649 by NAT_4:9; :: thesis:
9649 = (83 * 116) + 21 ;
hence not 83 divides 9649 by NAT_4:9; :: thesis:
9649 = (89 * 108) + 37 ;
hence not 89 divides 9649 by NAT_4:9; :: thesis:
9649 = (97 * 99) + 46 ;
hence not 97 divides 9649 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9649 & n is prime holds
not n divides 9649 by XPRIMET1:50;
hence 9649 is prime by NAT_4:14; :: thesis: