6421 = (2 * 3210) + 1 ;
hence not 2 divides 6421 by NAT_4:9; :: thesis:
6421 = (3 * 2140) + 1 ;
hence not 3 divides 6421 by NAT_4:9; :: thesis:
6421 = (5 * 1284) + 1 ;
hence not 5 divides 6421 by NAT_4:9; :: thesis:
6421 = (7 * 917) + 2 ;
hence not 7 divides 6421 by NAT_4:9; :: thesis:
6421 = (11 * 583) + 8 ;
hence not 11 divides 6421 by NAT_4:9; :: thesis:
6421 = (13 * 493) + 12 ;
hence not 13 divides 6421 by NAT_4:9; :: thesis:
6421 = (17 * 377) + 12 ;
hence not 17 divides 6421 by NAT_4:9; :: thesis:
6421 = (19 * 337) + 18 ;
hence not 19 divides 6421 by NAT_4:9; :: thesis:
6421 = (23 * 279) + 4 ;
hence not 23 divides 6421 by NAT_4:9; :: thesis:
6421 = (29 * 221) + 12 ;
hence not 29 divides 6421 by NAT_4:9; :: thesis:
6421 = (31 * 207) + 4 ;
hence not 31 divides 6421 by NAT_4:9; :: thesis:
6421 = (37 * 173) + 20 ;
hence not 37 divides 6421 by NAT_4:9; :: thesis:
6421 = (41 * 156) + 25 ;
hence not 41 divides 6421 by NAT_4:9; :: thesis:
6421 = (43 * 149) + 14 ;
hence not 43 divides 6421 by NAT_4:9; :: thesis:
6421 = (47 * 136) + 29 ;
hence not 47 divides 6421 by NAT_4:9; :: thesis:
6421 = (53 * 121) + 8 ;
hence not 53 divides 6421 by NAT_4:9; :: thesis:
6421 = (59 * 108) + 49 ;
hence not 59 divides 6421 by NAT_4:9; :: thesis:
6421 = (61 * 105) + 16 ;
hence not 61 divides 6421 by NAT_4:9; :: thesis:
6421 = (67 * 95) + 56 ;
hence not 67 divides 6421 by NAT_4:9; :: thesis:
6421 = (71 * 90) + 31 ;
hence not 71 divides 6421 by NAT_4:9; :: thesis:
6421 = (73 * 87) + 70 ;
hence not 73 divides 6421 by NAT_4:9; :: thesis:
6421 = (79 * 81) + 22 ;
hence not 79 divides 6421 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6421 & n is prime holds
not n divides 6421 by XPRIMET1:44;
hence 6421 is prime by NAT_4:14; :: thesis: