9973 = (2 * 4986) + 1 ;
hence not 2 divides 9973 by NAT_4:9; :: thesis:
9973 = (3 * 3324) + 1 ;
hence not 3 divides 9973 by NAT_4:9; :: thesis:
9973 = (5 * 1994) + 3 ;
hence not 5 divides 9973 by NAT_4:9; :: thesis:
9973 = (7 * 1424) + 5 ;
hence not 7 divides 9973 by NAT_4:9; :: thesis:
9973 = (11 * 906) + 7 ;
hence not 11 divides 9973 by NAT_4:9; :: thesis:
9973 = (13 * 767) + 2 ;
hence not 13 divides 9973 by NAT_4:9; :: thesis:
9973 = (17 * 586) + 11 ;
hence not 17 divides 9973 by NAT_4:9; :: thesis:
9973 = (19 * 524) + 17 ;
hence not 19 divides 9973 by NAT_4:9; :: thesis:
9973 = (23 * 433) + 14 ;
hence not 23 divides 9973 by NAT_4:9; :: thesis:
9973 = (29 * 343) + 26 ;
hence not 29 divides 9973 by NAT_4:9; :: thesis:
9973 = (31 * 321) + 22 ;
hence not 31 divides 9973 by NAT_4:9; :: thesis:
9973 = (37 * 269) + 20 ;
hence not 37 divides 9973 by NAT_4:9; :: thesis:
9973 = (41 * 243) + 10 ;
hence not 41 divides 9973 by NAT_4:9; :: thesis:
9973 = (43 * 231) + 40 ;
hence not 43 divides 9973 by NAT_4:9; :: thesis:
9973 = (47 * 212) + 9 ;
hence not 47 divides 9973 by NAT_4:9; :: thesis:
9973 = (53 * 188) + 9 ;
hence not 53 divides 9973 by NAT_4:9; :: thesis:
9973 = (59 * 169) + 2 ;
hence not 59 divides 9973 by NAT_4:9; :: thesis:
9973 = (61 * 163) + 30 ;
hence not 61 divides 9973 by NAT_4:9; :: thesis:
9973 = (67 * 148) + 57 ;
hence not 67 divides 9973 by NAT_4:9; :: thesis:
9973 = (71 * 140) + 33 ;
hence not 71 divides 9973 by NAT_4:9; :: thesis:
9973 = (73 * 136) + 45 ;
hence not 73 divides 9973 by NAT_4:9; :: thesis:
9973 = (79 * 126) + 19 ;
hence not 79 divides 9973 by NAT_4:9; :: thesis:
9973 = (83 * 120) + 13 ;
hence not 83 divides 9973 by NAT_4:9; :: thesis:
9973 = (89 * 112) + 5 ;
hence not 89 divides 9973 by NAT_4:9; :: thesis:
9973 = (97 * 102) + 79 ;
hence not 97 divides 9973 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9973 & n is prime holds
not n divides 9973 by XPRIMET1:50;
hence 9973 is prime by NAT_4:14; :: thesis: