9941 = (2 * 4970) + 1 ;
hence not 2 divides 9941 by NAT_4:9; :: thesis:
9941 = (3 * 3313) + 2 ;
hence not 3 divides 9941 by NAT_4:9; :: thesis:
9941 = (5 * 1988) + 1 ;
hence not 5 divides 9941 by NAT_4:9; :: thesis:
9941 = (7 * 1420) + 1 ;
hence not 7 divides 9941 by NAT_4:9; :: thesis:
9941 = (11 * 903) + 8 ;
hence not 11 divides 9941 by NAT_4:9; :: thesis:
9941 = (13 * 764) + 9 ;
hence not 13 divides 9941 by NAT_4:9; :: thesis:
9941 = (17 * 584) + 13 ;
hence not 17 divides 9941 by NAT_4:9; :: thesis:
9941 = (19 * 523) + 4 ;
hence not 19 divides 9941 by NAT_4:9; :: thesis:
9941 = (23 * 432) + 5 ;
hence not 23 divides 9941 by NAT_4:9; :: thesis:
9941 = (29 * 342) + 23 ;
hence not 29 divides 9941 by NAT_4:9; :: thesis:
9941 = (31 * 320) + 21 ;
hence not 31 divides 9941 by NAT_4:9; :: thesis:
9941 = (37 * 268) + 25 ;
hence not 37 divides 9941 by NAT_4:9; :: thesis:
9941 = (41 * 242) + 19 ;
hence not 41 divides 9941 by NAT_4:9; :: thesis:
9941 = (43 * 231) + 8 ;
hence not 43 divides 9941 by NAT_4:9; :: thesis:
9941 = (47 * 211) + 24 ;
hence not 47 divides 9941 by NAT_4:9; :: thesis:
9941 = (53 * 187) + 30 ;
hence not 53 divides 9941 by NAT_4:9; :: thesis:
9941 = (59 * 168) + 29 ;
hence not 59 divides 9941 by NAT_4:9; :: thesis:
9941 = (61 * 162) + 59 ;
hence not 61 divides 9941 by NAT_4:9; :: thesis:
9941 = (67 * 148) + 25 ;
hence not 67 divides 9941 by NAT_4:9; :: thesis:
9941 = (71 * 140) + 1 ;
hence not 71 divides 9941 by NAT_4:9; :: thesis:
9941 = (73 * 136) + 13 ;
hence not 73 divides 9941 by NAT_4:9; :: thesis:
9941 = (79 * 125) + 66 ;
hence not 79 divides 9941 by NAT_4:9; :: thesis:
9941 = (83 * 119) + 64 ;
hence not 83 divides 9941 by NAT_4:9; :: thesis:
9941 = (89 * 111) + 62 ;
hence not 89 divides 9941 by NAT_4:9; :: thesis:
9941 = (97 * 102) + 47 ;
hence not 97 divides 9941 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9941 & n is prime holds
not n divides 9941 by XPRIMET1:50;
hence 9941 is prime by NAT_4:14; :: thesis: