4943 = (2 * 2471) + 1 ;
hence not 2 divides 4943 by NAT_4:9; :: thesis:
4943 = (3 * 1647) + 2 ;
hence not 3 divides 4943 by NAT_4:9; :: thesis:
4943 = (5 * 988) + 3 ;
hence not 5 divides 4943 by NAT_4:9; :: thesis:
4943 = (7 * 706) + 1 ;
hence not 7 divides 4943 by NAT_4:9; :: thesis:
4943 = (11 * 449) + 4 ;
hence not 11 divides 4943 by NAT_4:9; :: thesis:
4943 = (13 * 380) + 3 ;
hence not 13 divides 4943 by NAT_4:9; :: thesis:
4943 = (17 * 290) + 13 ;
hence not 17 divides 4943 by NAT_4:9; :: thesis:
4943 = (19 * 260) + 3 ;
hence not 19 divides 4943 by NAT_4:9; :: thesis:
4943 = (23 * 214) + 21 ;
hence not 23 divides 4943 by NAT_4:9; :: thesis:
4943 = (29 * 170) + 13 ;
hence not 29 divides 4943 by NAT_4:9; :: thesis:
4943 = (31 * 159) + 14 ;
hence not 31 divides 4943 by NAT_4:9; :: thesis:
4943 = (37 * 133) + 22 ;
hence not 37 divides 4943 by NAT_4:9; :: thesis:
4943 = (41 * 120) + 23 ;
hence not 41 divides 4943 by NAT_4:9; :: thesis:
4943 = (43 * 114) + 41 ;
hence not 43 divides 4943 by NAT_4:9; :: thesis:
4943 = (47 * 105) + 8 ;
hence not 47 divides 4943 by NAT_4:9; :: thesis:
4943 = (53 * 93) + 14 ;
hence not 53 divides 4943 by NAT_4:9; :: thesis:
4943 = (59 * 83) + 46 ;
hence not 59 divides 4943 by NAT_4:9; :: thesis:
4943 = (61 * 81) + 2 ;
hence not 61 divides 4943 by NAT_4:9; :: thesis:
4943 = (67 * 73) + 52 ;
hence not 67 divides 4943 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4943 & n is prime holds
not n divides 4943 by XPRIMET1:38;
hence 4943 is prime by NAT_4:14; :: thesis: