5081 = (2 * 2540) + 1 ;
hence not 2 divides 5081 by NAT_4:9; :: thesis:
5081 = (3 * 1693) + 2 ;
hence not 3 divides 5081 by NAT_4:9; :: thesis:
5081 = (5 * 1016) + 1 ;
hence not 5 divides 5081 by NAT_4:9; :: thesis:
5081 = (7 * 725) + 6 ;
hence not 7 divides 5081 by NAT_4:9; :: thesis:
5081 = (11 * 461) + 10 ;
hence not 11 divides 5081 by NAT_4:9; :: thesis:
5081 = (13 * 390) + 11 ;
hence not 13 divides 5081 by NAT_4:9; :: thesis:
5081 = (17 * 298) + 15 ;
hence not 17 divides 5081 by NAT_4:9; :: thesis:
5081 = (19 * 267) + 8 ;
hence not 19 divides 5081 by NAT_4:9; :: thesis:
5081 = (23 * 220) + 21 ;
hence not 23 divides 5081 by NAT_4:9; :: thesis:
5081 = (29 * 175) + 6 ;
hence not 29 divides 5081 by NAT_4:9; :: thesis:
5081 = (31 * 163) + 28 ;
hence not 31 divides 5081 by NAT_4:9; :: thesis:
5081 = (37 * 137) + 12 ;
hence not 37 divides 5081 by NAT_4:9; :: thesis:
5081 = (41 * 123) + 38 ;
hence not 41 divides 5081 by NAT_4:9; :: thesis:
5081 = (43 * 118) + 7 ;
hence not 43 divides 5081 by NAT_4:9; :: thesis:
5081 = (47 * 108) + 5 ;
hence not 47 divides 5081 by NAT_4:9; :: thesis:
5081 = (53 * 95) + 46 ;
hence not 53 divides 5081 by NAT_4:9; :: thesis:
5081 = (59 * 86) + 7 ;
hence not 59 divides 5081 by NAT_4:9; :: thesis:
5081 = (61 * 83) + 18 ;
hence not 61 divides 5081 by NAT_4:9; :: thesis:
5081 = (67 * 75) + 56 ;
hence not 67 divides 5081 by NAT_4:9; :: thesis:
5081 = (71 * 71) + 40 ;
hence not 71 divides 5081 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5081 & n is prime holds
not n divides 5081 by XPRIMET1:40;
hence 5081 is prime by NAT_4:14; :: thesis: