3041 = (2 * 1520) + 1 ;
hence not 2 divides 3041 by NAT_4:9; :: thesis:
3041 = (3 * 1013) + 2 ;
hence not 3 divides 3041 by NAT_4:9; :: thesis:
3041 = (5 * 608) + 1 ;
hence not 5 divides 3041 by NAT_4:9; :: thesis:
3041 = (7 * 434) + 3 ;
hence not 7 divides 3041 by NAT_4:9; :: thesis:
3041 = (11 * 276) + 5 ;
hence not 11 divides 3041 by NAT_4:9; :: thesis:
3041 = (13 * 233) + 12 ;
hence not 13 divides 3041 by NAT_4:9; :: thesis:
3041 = (17 * 178) + 15 ;
hence not 17 divides 3041 by NAT_4:9; :: thesis:
3041 = (19 * 160) + 1 ;
hence not 19 divides 3041 by NAT_4:9; :: thesis:
3041 = (23 * 132) + 5 ;
hence not 23 divides 3041 by NAT_4:9; :: thesis:
3041 = (29 * 104) + 25 ;
hence not 29 divides 3041 by NAT_4:9; :: thesis:
3041 = (31 * 98) + 3 ;
hence not 31 divides 3041 by NAT_4:9; :: thesis:
3041 = (37 * 82) + 7 ;
hence not 37 divides 3041 by NAT_4:9; :: thesis:
3041 = (41 * 74) + 7 ;
hence not 41 divides 3041 by NAT_4:9; :: thesis:
3041 = (43 * 70) + 31 ;
hence not 43 divides 3041 by NAT_4:9; :: thesis:
3041 = (47 * 64) + 33 ;
hence not 47 divides 3041 by NAT_4:9; :: thesis:
3041 = (53 * 57) + 20 ;
hence not 53 divides 3041 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3041 & n is prime holds
not n divides 3041 by XPRIMET1:32;
hence 3041 is prime by NAT_4:14; :: thesis: