6389 = (2 * 3194) + 1 ;
hence not 2 divides 6389 by NAT_4:9; :: thesis:
6389 = (3 * 2129) + 2 ;
hence not 3 divides 6389 by NAT_4:9; :: thesis:
6389 = (5 * 1277) + 4 ;
hence not 5 divides 6389 by NAT_4:9; :: thesis:
6389 = (7 * 912) + 5 ;
hence not 7 divides 6389 by NAT_4:9; :: thesis:
6389 = (11 * 580) + 9 ;
hence not 11 divides 6389 by NAT_4:9; :: thesis:
6389 = (13 * 491) + 6 ;
hence not 13 divides 6389 by NAT_4:9; :: thesis:
6389 = (17 * 375) + 14 ;
hence not 17 divides 6389 by NAT_4:9; :: thesis:
6389 = (19 * 336) + 5 ;
hence not 19 divides 6389 by NAT_4:9; :: thesis:
6389 = (23 * 277) + 18 ;
hence not 23 divides 6389 by NAT_4:9; :: thesis:
6389 = (29 * 220) + 9 ;
hence not 29 divides 6389 by NAT_4:9; :: thesis:
6389 = (31 * 206) + 3 ;
hence not 31 divides 6389 by NAT_4:9; :: thesis:
6389 = (37 * 172) + 25 ;
hence not 37 divides 6389 by NAT_4:9; :: thesis:
6389 = (41 * 155) + 34 ;
hence not 41 divides 6389 by NAT_4:9; :: thesis:
6389 = (43 * 148) + 25 ;
hence not 43 divides 6389 by NAT_4:9; :: thesis:
6389 = (47 * 135) + 44 ;
hence not 47 divides 6389 by NAT_4:9; :: thesis:
6389 = (53 * 120) + 29 ;
hence not 53 divides 6389 by NAT_4:9; :: thesis:
6389 = (59 * 108) + 17 ;
hence not 59 divides 6389 by NAT_4:9; :: thesis:
6389 = (61 * 104) + 45 ;
hence not 61 divides 6389 by NAT_4:9; :: thesis:
6389 = (67 * 95) + 24 ;
hence not 67 divides 6389 by NAT_4:9; :: thesis:
6389 = (71 * 89) + 70 ;
hence not 71 divides 6389 by NAT_4:9; :: thesis:
6389 = (73 * 87) + 38 ;
hence not 73 divides 6389 by NAT_4:9; :: thesis:
6389 = (79 * 80) + 69 ;
hence not 79 divides 6389 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6389 & n is prime holds
not n divides 6389 by XPRIMET1:44;
hence 6389 is prime by NAT_4:14; :: thesis: