7741 = (2 * 3870) + 1 ;
hence not 2 divides 7741 by NAT_4:9; :: thesis:
7741 = (3 * 2580) + 1 ;
hence not 3 divides 7741 by NAT_4:9; :: thesis:
7741 = (5 * 1548) + 1 ;
hence not 5 divides 7741 by NAT_4:9; :: thesis:
7741 = (7 * 1105) + 6 ;
hence not 7 divides 7741 by NAT_4:9; :: thesis:
7741 = (11 * 703) + 8 ;
hence not 11 divides 7741 by NAT_4:9; :: thesis:
7741 = (13 * 595) + 6 ;
hence not 13 divides 7741 by NAT_4:9; :: thesis:
7741 = (17 * 455) + 6 ;
hence not 17 divides 7741 by NAT_4:9; :: thesis:
7741 = (19 * 407) + 8 ;
hence not 19 divides 7741 by NAT_4:9; :: thesis:
7741 = (23 * 336) + 13 ;
hence not 23 divides 7741 by NAT_4:9; :: thesis:
7741 = (29 * 266) + 27 ;
hence not 29 divides 7741 by NAT_4:9; :: thesis:
7741 = (31 * 249) + 22 ;
hence not 31 divides 7741 by NAT_4:9; :: thesis:
7741 = (37 * 209) + 8 ;
hence not 37 divides 7741 by NAT_4:9; :: thesis:
7741 = (41 * 188) + 33 ;
hence not 41 divides 7741 by NAT_4:9; :: thesis:
7741 = (43 * 180) + 1 ;
hence not 43 divides 7741 by NAT_4:9; :: thesis:
7741 = (47 * 164) + 33 ;
hence not 47 divides 7741 by NAT_4:9; :: thesis:
7741 = (53 * 146) + 3 ;
hence not 53 divides 7741 by NAT_4:9; :: thesis:
7741 = (59 * 131) + 12 ;
hence not 59 divides 7741 by NAT_4:9; :: thesis:
7741 = (61 * 126) + 55 ;
hence not 61 divides 7741 by NAT_4:9; :: thesis:
7741 = (67 * 115) + 36 ;
hence not 67 divides 7741 by NAT_4:9; :: thesis:
7741 = (71 * 109) + 2 ;
hence not 71 divides 7741 by NAT_4:9; :: thesis:
7741 = (73 * 106) + 3 ;
hence not 73 divides 7741 by NAT_4:9; :: thesis:
7741 = (79 * 97) + 78 ;
hence not 79 divides 7741 by NAT_4:9; :: thesis:
7741 = (83 * 93) + 22 ;
hence not 83 divides 7741 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7741 & n is prime holds
not n divides 7741 by XPRIMET1:46;
hence 7741 is prime by NAT_4:14; :: thesis: