4111 = (2 * 2055) + 1 ;
hence not 2 divides 4111 by NAT_4:9; :: thesis:
4111 = (3 * 1370) + 1 ;
hence not 3 divides 4111 by NAT_4:9; :: thesis:
4111 = (5 * 822) + 1 ;
hence not 5 divides 4111 by NAT_4:9; :: thesis:
4111 = (7 * 587) + 2 ;
hence not 7 divides 4111 by NAT_4:9; :: thesis:
4111 = (11 * 373) + 8 ;
hence not 11 divides 4111 by NAT_4:9; :: thesis:
4111 = (13 * 316) + 3 ;
hence not 13 divides 4111 by NAT_4:9; :: thesis:
4111 = (17 * 241) + 14 ;
hence not 17 divides 4111 by NAT_4:9; :: thesis:
4111 = (19 * 216) + 7 ;
hence not 19 divides 4111 by NAT_4:9; :: thesis:
4111 = (23 * 178) + 17 ;
hence not 23 divides 4111 by NAT_4:9; :: thesis:
4111 = (29 * 141) + 22 ;
hence not 29 divides 4111 by NAT_4:9; :: thesis:
4111 = (31 * 132) + 19 ;
hence not 31 divides 4111 by NAT_4:9; :: thesis:
4111 = (37 * 111) + 4 ;
hence not 37 divides 4111 by NAT_4:9; :: thesis:
4111 = (41 * 100) + 11 ;
hence not 41 divides 4111 by NAT_4:9; :: thesis:
4111 = (43 * 95) + 26 ;
hence not 43 divides 4111 by NAT_4:9; :: thesis:
4111 = (47 * 87) + 22 ;
hence not 47 divides 4111 by NAT_4:9; :: thesis:
4111 = (53 * 77) + 30 ;
hence not 53 divides 4111 by NAT_4:9; :: thesis:
4111 = (59 * 69) + 40 ;
hence not 59 divides 4111 by NAT_4:9; :: thesis:
4111 = (61 * 67) + 24 ;
hence not 61 divides 4111 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 4111 & n is prime holds
not n divides 4111 by XPRIMET1:36;
hence 4111 is prime by NAT_4:14; :: thesis: