6451 = (2 * 3225) + 1 ;
hence not 2 divides 6451 by NAT_4:9; :: thesis:
6451 = (3 * 2150) + 1 ;
hence not 3 divides 6451 by NAT_4:9; :: thesis:
6451 = (5 * 1290) + 1 ;
hence not 5 divides 6451 by NAT_4:9; :: thesis:
6451 = (7 * 921) + 4 ;
hence not 7 divides 6451 by NAT_4:9; :: thesis:
6451 = (11 * 586) + 5 ;
hence not 11 divides 6451 by NAT_4:9; :: thesis:
6451 = (13 * 496) + 3 ;
hence not 13 divides 6451 by NAT_4:9; :: thesis:
6451 = (17 * 379) + 8 ;
hence not 17 divides 6451 by NAT_4:9; :: thesis:
6451 = (19 * 339) + 10 ;
hence not 19 divides 6451 by NAT_4:9; :: thesis:
6451 = (23 * 280) + 11 ;
hence not 23 divides 6451 by NAT_4:9; :: thesis:
6451 = (29 * 222) + 13 ;
hence not 29 divides 6451 by NAT_4:9; :: thesis:
6451 = (31 * 208) + 3 ;
hence not 31 divides 6451 by NAT_4:9; :: thesis:
6451 = (37 * 174) + 13 ;
hence not 37 divides 6451 by NAT_4:9; :: thesis:
6451 = (41 * 157) + 14 ;
hence not 41 divides 6451 by NAT_4:9; :: thesis:
6451 = (43 * 150) + 1 ;
hence not 43 divides 6451 by NAT_4:9; :: thesis:
6451 = (47 * 137) + 12 ;
hence not 47 divides 6451 by NAT_4:9; :: thesis:
6451 = (53 * 121) + 38 ;
hence not 53 divides 6451 by NAT_4:9; :: thesis:
6451 = (59 * 109) + 20 ;
hence not 59 divides 6451 by NAT_4:9; :: thesis:
6451 = (61 * 105) + 46 ;
hence not 61 divides 6451 by NAT_4:9; :: thesis:
6451 = (67 * 96) + 19 ;
hence not 67 divides 6451 by NAT_4:9; :: thesis:
6451 = (71 * 90) + 61 ;
hence not 71 divides 6451 by NAT_4:9; :: thesis:
6451 = (73 * 88) + 27 ;
hence not 73 divides 6451 by NAT_4:9; :: thesis:
6451 = (79 * 81) + 52 ;
hence not 79 divides 6451 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6451 & n is prime holds
not n divides 6451 by XPRIMET1:44;
hence 6451 is prime by NAT_4:14; :: thesis: