9871 = (2 * 4935) + 1 ;
hence not 2 divides 9871 by NAT_4:9; :: thesis:
9871 = (3 * 3290) + 1 ;
hence not 3 divides 9871 by NAT_4:9; :: thesis:
9871 = (5 * 1974) + 1 ;
hence not 5 divides 9871 by NAT_4:9; :: thesis:
9871 = (7 * 1410) + 1 ;
hence not 7 divides 9871 by NAT_4:9; :: thesis:
9871 = (11 * 897) + 4 ;
hence not 11 divides 9871 by NAT_4:9; :: thesis:
9871 = (13 * 759) + 4 ;
hence not 13 divides 9871 by NAT_4:9; :: thesis:
9871 = (17 * 580) + 11 ;
hence not 17 divides 9871 by NAT_4:9; :: thesis:
9871 = (19 * 519) + 10 ;
hence not 19 divides 9871 by NAT_4:9; :: thesis:
9871 = (23 * 429) + 4 ;
hence not 23 divides 9871 by NAT_4:9; :: thesis:
9871 = (29 * 340) + 11 ;
hence not 29 divides 9871 by NAT_4:9; :: thesis:
9871 = (31 * 318) + 13 ;
hence not 31 divides 9871 by NAT_4:9; :: thesis:
9871 = (37 * 266) + 29 ;
hence not 37 divides 9871 by NAT_4:9; :: thesis:
9871 = (41 * 240) + 31 ;
hence not 41 divides 9871 by NAT_4:9; :: thesis:
9871 = (43 * 229) + 24 ;
hence not 43 divides 9871 by NAT_4:9; :: thesis:
9871 = (47 * 210) + 1 ;
hence not 47 divides 9871 by NAT_4:9; :: thesis:
9871 = (53 * 186) + 13 ;
hence not 53 divides 9871 by NAT_4:9; :: thesis:
9871 = (59 * 167) + 18 ;
hence not 59 divides 9871 by NAT_4:9; :: thesis:
9871 = (61 * 161) + 50 ;
hence not 61 divides 9871 by NAT_4:9; :: thesis:
9871 = (67 * 147) + 22 ;
hence not 67 divides 9871 by NAT_4:9; :: thesis:
9871 = (71 * 139) + 2 ;
hence not 71 divides 9871 by NAT_4:9; :: thesis:
9871 = (73 * 135) + 16 ;
hence not 73 divides 9871 by NAT_4:9; :: thesis:
9871 = (79 * 124) + 75 ;
hence not 79 divides 9871 by NAT_4:9; :: thesis:
9871 = (83 * 118) + 77 ;
hence not 83 divides 9871 by NAT_4:9; :: thesis:
9871 = (89 * 110) + 81 ;
hence not 89 divides 9871 by NAT_4:9; :: thesis:
9871 = (97 * 101) + 74 ;
hence not 97 divides 9871 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9871 & n is prime holds
not n divides 9871 by XPRIMET1:50;
hence 9871 is prime by NAT_4:14; :: thesis: