3049 = (2 * 1524) + 1 ;
hence not 2 divides 3049 by NAT_4:9; :: thesis:
3049 = (3 * 1016) + 1 ;
hence not 3 divides 3049 by NAT_4:9; :: thesis:
3049 = (5 * 609) + 4 ;
hence not 5 divides 3049 by NAT_4:9; :: thesis:
3049 = (7 * 435) + 4 ;
hence not 7 divides 3049 by NAT_4:9; :: thesis:
3049 = (11 * 277) + 2 ;
hence not 11 divides 3049 by NAT_4:9; :: thesis:
3049 = (13 * 234) + 7 ;
hence not 13 divides 3049 by NAT_4:9; :: thesis:
3049 = (17 * 179) + 6 ;
hence not 17 divides 3049 by NAT_4:9; :: thesis:
3049 = (19 * 160) + 9 ;
hence not 19 divides 3049 by NAT_4:9; :: thesis:
3049 = (23 * 132) + 13 ;
hence not 23 divides 3049 by NAT_4:9; :: thesis:
3049 = (29 * 105) + 4 ;
hence not 29 divides 3049 by NAT_4:9; :: thesis:
3049 = (31 * 98) + 11 ;
hence not 31 divides 3049 by NAT_4:9; :: thesis:
3049 = (37 * 82) + 15 ;
hence not 37 divides 3049 by NAT_4:9; :: thesis:
3049 = (41 * 74) + 15 ;
hence not 41 divides 3049 by NAT_4:9; :: thesis:
3049 = (43 * 70) + 39 ;
hence not 43 divides 3049 by NAT_4:9; :: thesis:
3049 = (47 * 64) + 41 ;
hence not 47 divides 3049 by NAT_4:9; :: thesis:
3049 = (53 * 57) + 28 ;
hence not 53 divides 3049 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3049 & n is prime holds
not n divides 3049 by XPRIMET1:32;
hence 3049 is prime by NAT_4:14; :: thesis: