3061 = (2 * 1530) + 1 ;
hence not 2 divides 3061 by NAT_4:9; :: thesis:
3061 = (3 * 1020) + 1 ;
hence not 3 divides 3061 by NAT_4:9; :: thesis:
3061 = (5 * 612) + 1 ;
hence not 5 divides 3061 by NAT_4:9; :: thesis:
3061 = (7 * 437) + 2 ;
hence not 7 divides 3061 by NAT_4:9; :: thesis:
3061 = (11 * 278) + 3 ;
hence not 11 divides 3061 by NAT_4:9; :: thesis:
3061 = (13 * 235) + 6 ;
hence not 13 divides 3061 by NAT_4:9; :: thesis:
3061 = (17 * 180) + 1 ;
hence not 17 divides 3061 by NAT_4:9; :: thesis:
3061 = (19 * 161) + 2 ;
hence not 19 divides 3061 by NAT_4:9; :: thesis:
3061 = (23 * 133) + 2 ;
hence not 23 divides 3061 by NAT_4:9; :: thesis:
3061 = (29 * 105) + 16 ;
hence not 29 divides 3061 by NAT_4:9; :: thesis:
3061 = (31 * 98) + 23 ;
hence not 31 divides 3061 by NAT_4:9; :: thesis:
3061 = (37 * 82) + 27 ;
hence not 37 divides 3061 by NAT_4:9; :: thesis:
3061 = (41 * 74) + 27 ;
hence not 41 divides 3061 by NAT_4:9; :: thesis:
3061 = (43 * 71) + 8 ;
hence not 43 divides 3061 by NAT_4:9; :: thesis:
3061 = (47 * 65) + 6 ;
hence not 47 divides 3061 by NAT_4:9; :: thesis:
3061 = (53 * 57) + 40 ;
hence not 53 divides 3061 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3061 & n is prime holds
not n divides 3061 by XPRIMET1:32;
hence 3061 is prime by NAT_4:14; :: thesis: