6367 = (2 * 3183) + 1 ;
hence not 2 divides 6367 by NAT_4:9; :: thesis:
6367 = (3 * 2122) + 1 ;
hence not 3 divides 6367 by NAT_4:9; :: thesis:
6367 = (5 * 1273) + 2 ;
hence not 5 divides 6367 by NAT_4:9; :: thesis:
6367 = (7 * 909) + 4 ;
hence not 7 divides 6367 by NAT_4:9; :: thesis:
6367 = (11 * 578) + 9 ;
hence not 11 divides 6367 by NAT_4:9; :: thesis:
6367 = (13 * 489) + 10 ;
hence not 13 divides 6367 by NAT_4:9; :: thesis:
6367 = (17 * 374) + 9 ;
hence not 17 divides 6367 by NAT_4:9; :: thesis:
6367 = (19 * 335) + 2 ;
hence not 19 divides 6367 by NAT_4:9; :: thesis:
6367 = (23 * 276) + 19 ;
hence not 23 divides 6367 by NAT_4:9; :: thesis:
6367 = (29 * 219) + 16 ;
hence not 29 divides 6367 by NAT_4:9; :: thesis:
6367 = (31 * 205) + 12 ;
hence not 31 divides 6367 by NAT_4:9; :: thesis:
6367 = (37 * 172) + 3 ;
hence not 37 divides 6367 by NAT_4:9; :: thesis:
6367 = (41 * 155) + 12 ;
hence not 41 divides 6367 by NAT_4:9; :: thesis:
6367 = (43 * 148) + 3 ;
hence not 43 divides 6367 by NAT_4:9; :: thesis:
6367 = (47 * 135) + 22 ;
hence not 47 divides 6367 by NAT_4:9; :: thesis:
6367 = (53 * 120) + 7 ;
hence not 53 divides 6367 by NAT_4:9; :: thesis:
6367 = (59 * 107) + 54 ;
hence not 59 divides 6367 by NAT_4:9; :: thesis:
6367 = (61 * 104) + 23 ;
hence not 61 divides 6367 by NAT_4:9; :: thesis:
6367 = (67 * 95) + 2 ;
hence not 67 divides 6367 by NAT_4:9; :: thesis:
6367 = (71 * 89) + 48 ;
hence not 71 divides 6367 by NAT_4:9; :: thesis:
6367 = (73 * 87) + 16 ;
hence not 73 divides 6367 by NAT_4:9; :: thesis:
6367 = (79 * 80) + 47 ;
hence not 79 divides 6367 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6367 & n is prime holds
not n divides 6367 by XPRIMET1:44;
hence 6367 is prime by NAT_4:14; :: thesis: