6547 = (2 * 3273) + 1 ;
hence not 2 divides 6547 by NAT_4:9; :: thesis:
6547 = (3 * 2182) + 1 ;
hence not 3 divides 6547 by NAT_4:9; :: thesis:
6547 = (5 * 1309) + 2 ;
hence not 5 divides 6547 by NAT_4:9; :: thesis:
6547 = (7 * 935) + 2 ;
hence not 7 divides 6547 by NAT_4:9; :: thesis:
6547 = (11 * 595) + 2 ;
hence not 11 divides 6547 by NAT_4:9; :: thesis:
6547 = (13 * 503) + 8 ;
hence not 13 divides 6547 by NAT_4:9; :: thesis:
6547 = (17 * 385) + 2 ;
hence not 17 divides 6547 by NAT_4:9; :: thesis:
6547 = (19 * 344) + 11 ;
hence not 19 divides 6547 by NAT_4:9; :: thesis:
6547 = (23 * 284) + 15 ;
hence not 23 divides 6547 by NAT_4:9; :: thesis:
6547 = (29 * 225) + 22 ;
hence not 29 divides 6547 by NAT_4:9; :: thesis:
6547 = (31 * 211) + 6 ;
hence not 31 divides 6547 by NAT_4:9; :: thesis:
6547 = (37 * 176) + 35 ;
hence not 37 divides 6547 by NAT_4:9; :: thesis:
6547 = (41 * 159) + 28 ;
hence not 41 divides 6547 by NAT_4:9; :: thesis:
6547 = (43 * 152) + 11 ;
hence not 43 divides 6547 by NAT_4:9; :: thesis:
6547 = (47 * 139) + 14 ;
hence not 47 divides 6547 by NAT_4:9; :: thesis:
6547 = (53 * 123) + 28 ;
hence not 53 divides 6547 by NAT_4:9; :: thesis:
6547 = (59 * 110) + 57 ;
hence not 59 divides 6547 by NAT_4:9; :: thesis:
6547 = (61 * 107) + 20 ;
hence not 61 divides 6547 by NAT_4:9; :: thesis:
6547 = (67 * 97) + 48 ;
hence not 67 divides 6547 by NAT_4:9; :: thesis:
6547 = (71 * 92) + 15 ;
hence not 71 divides 6547 by NAT_4:9; :: thesis:
6547 = (73 * 89) + 50 ;
hence not 73 divides 6547 by NAT_4:9; :: thesis:
6547 = (79 * 82) + 69 ;
hence not 79 divides 6547 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6547 & n is prime holds
not n divides 6547 by XPRIMET1:44;
hence 6547 is prime by NAT_4:14; :: thesis: