3853 = (2 * 1926) + 1 ;
hence not 2 divides 3853 by NAT_4:9; :: thesis:
3853 = (3 * 1284) + 1 ;
hence not 3 divides 3853 by NAT_4:9; :: thesis:
3853 = (5 * 770) + 3 ;
hence not 5 divides 3853 by NAT_4:9; :: thesis:
3853 = (7 * 550) + 3 ;
hence not 7 divides 3853 by NAT_4:9; :: thesis:
3853 = (11 * 350) + 3 ;
hence not 11 divides 3853 by NAT_4:9; :: thesis:
3853 = (13 * 296) + 5 ;
hence not 13 divides 3853 by NAT_4:9; :: thesis:
3853 = (17 * 226) + 11 ;
hence not 17 divides 3853 by NAT_4:9; :: thesis:
3853 = (19 * 202) + 15 ;
hence not 19 divides 3853 by NAT_4:9; :: thesis:
3853 = (23 * 167) + 12 ;
hence not 23 divides 3853 by NAT_4:9; :: thesis:
3853 = (29 * 132) + 25 ;
hence not 29 divides 3853 by NAT_4:9; :: thesis:
3853 = (31 * 124) + 9 ;
hence not 31 divides 3853 by NAT_4:9; :: thesis:
3853 = (37 * 104) + 5 ;
hence not 37 divides 3853 by NAT_4:9; :: thesis:
3853 = (41 * 93) + 40 ;
hence not 41 divides 3853 by NAT_4:9; :: thesis:
3853 = (43 * 89) + 26 ;
hence not 43 divides 3853 by NAT_4:9; :: thesis:
3853 = (47 * 81) + 46 ;
hence not 47 divides 3853 by NAT_4:9; :: thesis:
3853 = (53 * 72) + 37 ;
hence not 53 divides 3853 by NAT_4:9; :: thesis:
3853 = (59 * 65) + 18 ;
hence not 59 divides 3853 by NAT_4:9; :: thesis:
3853 = (61 * 63) + 10 ;
hence not 61 divides 3853 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3853 & n is prime holds
not n divides 3853 by XPRIMET1:36;
hence 3853 is prime by NAT_4:14; :: thesis: