6427 = (2 * 3213) + 1 ;
hence not 2 divides 6427 by NAT_4:9; :: thesis:
6427 = (3 * 2142) + 1 ;
hence not 3 divides 6427 by NAT_4:9; :: thesis:
6427 = (5 * 1285) + 2 ;
hence not 5 divides 6427 by NAT_4:9; :: thesis:
6427 = (7 * 918) + 1 ;
hence not 7 divides 6427 by NAT_4:9; :: thesis:
6427 = (11 * 584) + 3 ;
hence not 11 divides 6427 by NAT_4:9; :: thesis:
6427 = (13 * 494) + 5 ;
hence not 13 divides 6427 by NAT_4:9; :: thesis:
6427 = (17 * 378) + 1 ;
hence not 17 divides 6427 by NAT_4:9; :: thesis:
6427 = (19 * 338) + 5 ;
hence not 19 divides 6427 by NAT_4:9; :: thesis:
6427 = (23 * 279) + 10 ;
hence not 23 divides 6427 by NAT_4:9; :: thesis:
6427 = (29 * 221) + 18 ;
hence not 29 divides 6427 by NAT_4:9; :: thesis:
6427 = (31 * 207) + 10 ;
hence not 31 divides 6427 by NAT_4:9; :: thesis:
6427 = (37 * 173) + 26 ;
hence not 37 divides 6427 by NAT_4:9; :: thesis:
6427 = (41 * 156) + 31 ;
hence not 41 divides 6427 by NAT_4:9; :: thesis:
6427 = (43 * 149) + 20 ;
hence not 43 divides 6427 by NAT_4:9; :: thesis:
6427 = (47 * 136) + 35 ;
hence not 47 divides 6427 by NAT_4:9; :: thesis:
6427 = (53 * 121) + 14 ;
hence not 53 divides 6427 by NAT_4:9; :: thesis:
6427 = (59 * 108) + 55 ;
hence not 59 divides 6427 by NAT_4:9; :: thesis:
6427 = (61 * 105) + 22 ;
hence not 61 divides 6427 by NAT_4:9; :: thesis:
6427 = (67 * 95) + 62 ;
hence not 67 divides 6427 by NAT_4:9; :: thesis:
6427 = (71 * 90) + 37 ;
hence not 71 divides 6427 by NAT_4:9; :: thesis:
6427 = (73 * 88) + 3 ;
hence not 73 divides 6427 by NAT_4:9; :: thesis:
6427 = (79 * 81) + 28 ;
hence not 79 divides 6427 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6427 & n is prime holds
not n divides 6427 by XPRIMET1:44;
hence 6427 is prime by NAT_4:14; :: thesis: