6397 = (2 * 3198) + 1 ;
hence not 2 divides 6397 by NAT_4:9; :: thesis:
6397 = (3 * 2132) + 1 ;
hence not 3 divides 6397 by NAT_4:9; :: thesis:
6397 = (5 * 1279) + 2 ;
hence not 5 divides 6397 by NAT_4:9; :: thesis:
6397 = (7 * 913) + 6 ;
hence not 7 divides 6397 by NAT_4:9; :: thesis:
6397 = (11 * 581) + 6 ;
hence not 11 divides 6397 by NAT_4:9; :: thesis:
6397 = (13 * 492) + 1 ;
hence not 13 divides 6397 by NAT_4:9; :: thesis:
6397 = (17 * 376) + 5 ;
hence not 17 divides 6397 by NAT_4:9; :: thesis:
6397 = (19 * 336) + 13 ;
hence not 19 divides 6397 by NAT_4:9; :: thesis:
6397 = (23 * 278) + 3 ;
hence not 23 divides 6397 by NAT_4:9; :: thesis:
6397 = (29 * 220) + 17 ;
hence not 29 divides 6397 by NAT_4:9; :: thesis:
6397 = (31 * 206) + 11 ;
hence not 31 divides 6397 by NAT_4:9; :: thesis:
6397 = (37 * 172) + 33 ;
hence not 37 divides 6397 by NAT_4:9; :: thesis:
6397 = (41 * 156) + 1 ;
hence not 41 divides 6397 by NAT_4:9; :: thesis:
6397 = (43 * 148) + 33 ;
hence not 43 divides 6397 by NAT_4:9; :: thesis:
6397 = (47 * 136) + 5 ;
hence not 47 divides 6397 by NAT_4:9; :: thesis:
6397 = (53 * 120) + 37 ;
hence not 53 divides 6397 by NAT_4:9; :: thesis:
6397 = (59 * 108) + 25 ;
hence not 59 divides 6397 by NAT_4:9; :: thesis:
6397 = (61 * 104) + 53 ;
hence not 61 divides 6397 by NAT_4:9; :: thesis:
6397 = (67 * 95) + 32 ;
hence not 67 divides 6397 by NAT_4:9; :: thesis:
6397 = (71 * 90) + 7 ;
hence not 71 divides 6397 by NAT_4:9; :: thesis:
6397 = (73 * 87) + 46 ;
hence not 73 divides 6397 by NAT_4:9; :: thesis:
6397 = (79 * 80) + 77 ;
hence not 79 divides 6397 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6397 & n is prime holds
not n divides 6397 by XPRIMET1:44;
hence 6397 is prime by NAT_4:14; :: thesis: