3823 = (2 * 1911) + 1 ;
hence not 2 divides 3823 by NAT_4:9; :: thesis:
3823 = (3 * 1274) + 1 ;
hence not 3 divides 3823 by NAT_4:9; :: thesis:
3823 = (5 * 764) + 3 ;
hence not 5 divides 3823 by NAT_4:9; :: thesis:
3823 = (7 * 546) + 1 ;
hence not 7 divides 3823 by NAT_4:9; :: thesis:
3823 = (11 * 347) + 6 ;
hence not 11 divides 3823 by NAT_4:9; :: thesis:
3823 = (13 * 294) + 1 ;
hence not 13 divides 3823 by NAT_4:9; :: thesis:
3823 = (17 * 224) + 15 ;
hence not 17 divides 3823 by NAT_4:9; :: thesis:
3823 = (19 * 201) + 4 ;
hence not 19 divides 3823 by NAT_4:9; :: thesis:
3823 = (23 * 166) + 5 ;
hence not 23 divides 3823 by NAT_4:9; :: thesis:
3823 = (29 * 131) + 24 ;
hence not 29 divides 3823 by NAT_4:9; :: thesis:
3823 = (31 * 123) + 10 ;
hence not 31 divides 3823 by NAT_4:9; :: thesis:
3823 = (37 * 103) + 12 ;
hence not 37 divides 3823 by NAT_4:9; :: thesis:
3823 = (41 * 93) + 10 ;
hence not 41 divides 3823 by NAT_4:9; :: thesis:
3823 = (43 * 88) + 39 ;
hence not 43 divides 3823 by NAT_4:9; :: thesis:
3823 = (47 * 81) + 16 ;
hence not 47 divides 3823 by NAT_4:9; :: thesis:
3823 = (53 * 72) + 7 ;
hence not 53 divides 3823 by NAT_4:9; :: thesis:
3823 = (59 * 64) + 47 ;
hence not 59 divides 3823 by NAT_4:9; :: thesis:
3823 = (61 * 62) + 41 ;
hence not 61 divides 3823 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3823 & n is prime holds
not n divides 3823 by XPRIMET1:36;
hence 3823 is prime by NAT_4:14; :: thesis: