4049 = (2 * 2024) + 1 ;
hence not 2 divides 4049 by NAT_4:9; :: thesis:
4049 = (3 * 1349) + 2 ;
hence not 3 divides 4049 by NAT_4:9; :: thesis:
4049 = (5 * 809) + 4 ;
hence not 5 divides 4049 by NAT_4:9; :: thesis:
4049 = (7 * 578) + 3 ;
hence not 7 divides 4049 by NAT_4:9; :: thesis:
4049 = (11 * 368) + 1 ;
hence not 11 divides 4049 by NAT_4:9; :: thesis:
4049 = (13 * 311) + 6 ;
hence not 13 divides 4049 by NAT_4:9; :: thesis:
4049 = (17 * 238) + 3 ;
hence not 17 divides 4049 by NAT_4:9; :: thesis:
4049 = (19 * 213) + 2 ;
hence not 19 divides 4049 by NAT_4:9; :: thesis:
4049 = (23 * 176) + 1 ;
hence not 23 divides 4049 by NAT_4:9; :: thesis:
4049 = (29 * 139) + 18 ;
hence not 29 divides 4049 by NAT_4:9; :: thesis:
4049 = (31 * 130) + 19 ;
hence not 31 divides 4049 by NAT_4:9; :: thesis:
4049 = (37 * 109) + 16 ;
hence not 37 divides 4049 by NAT_4:9; :: thesis:
4049 = (41 * 98) + 31 ;
hence not 41 divides 4049 by NAT_4:9; :: thesis:
4049 = (43 * 94) + 7 ;
hence not 43 divides 4049 by NAT_4:9; :: thesis:
4049 = (47 * 86) + 7 ;
hence not 47 divides 4049 by NAT_4:9; :: thesis:
4049 = (53 * 76) + 21 ;
hence not 53 divides 4049 by NAT_4:9; :: thesis:
4049 = (59 * 68) + 37 ;
hence not 59 divides 4049 by NAT_4:9; :: thesis:
4049 = (61 * 66) + 23 ;
hence not 61 divides 4049 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4049 & n is prime holds
not n divides 4049 by XPRIMET1:36;
hence 4049 is prime by NAT_4:14; :: thesis: