7993 = (2 * 3996) + 1 ;
hence not 2 divides 7993 by NAT_4:9; :: thesis:
7993 = (3 * 2664) + 1 ;
hence not 3 divides 7993 by NAT_4:9; :: thesis:
7993 = (5 * 1598) + 3 ;
hence not 5 divides 7993 by NAT_4:9; :: thesis:
7993 = (7 * 1141) + 6 ;
hence not 7 divides 7993 by NAT_4:9; :: thesis:
7993 = (11 * 726) + 7 ;
hence not 11 divides 7993 by NAT_4:9; :: thesis:
7993 = (13 * 614) + 11 ;
hence not 13 divides 7993 by NAT_4:9; :: thesis:
7993 = (17 * 470) + 3 ;
hence not 17 divides 7993 by NAT_4:9; :: thesis:
7993 = (19 * 420) + 13 ;
hence not 19 divides 7993 by NAT_4:9; :: thesis:
7993 = (23 * 347) + 12 ;
hence not 23 divides 7993 by NAT_4:9; :: thesis:
7993 = (29 * 275) + 18 ;
hence not 29 divides 7993 by NAT_4:9; :: thesis:
7993 = (31 * 257) + 26 ;
hence not 31 divides 7993 by NAT_4:9; :: thesis:
7993 = (37 * 216) + 1 ;
hence not 37 divides 7993 by NAT_4:9; :: thesis:
7993 = (41 * 194) + 39 ;
hence not 41 divides 7993 by NAT_4:9; :: thesis:
7993 = (43 * 185) + 38 ;
hence not 43 divides 7993 by NAT_4:9; :: thesis:
7993 = (47 * 170) + 3 ;
hence not 47 divides 7993 by NAT_4:9; :: thesis:
7993 = (53 * 150) + 43 ;
hence not 53 divides 7993 by NAT_4:9; :: thesis:
7993 = (59 * 135) + 28 ;
hence not 59 divides 7993 by NAT_4:9; :: thesis:
7993 = (61 * 131) + 2 ;
hence not 61 divides 7993 by NAT_4:9; :: thesis:
7993 = (67 * 119) + 20 ;
hence not 67 divides 7993 by NAT_4:9; :: thesis:
7993 = (71 * 112) + 41 ;
hence not 71 divides 7993 by NAT_4:9; :: thesis:
7993 = (73 * 109) + 36 ;
hence not 73 divides 7993 by NAT_4:9; :: thesis:
7993 = (79 * 101) + 14 ;
hence not 79 divides 7993 by NAT_4:9; :: thesis:
7993 = (83 * 96) + 25 ;
hence not 83 divides 7993 by NAT_4:9; :: thesis:
7993 = (89 * 89) + 72 ;
hence not 89 divides 7993 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7993 & n is prime holds
not n divides 7993 by XPRIMET1:48;
hence 7993 is prime by NAT_4:14; :: thesis: