547 = (2 * 273) + 1 ;
hence not 2 divides 547 by NAT_4:9; :: thesis:
547 = (3 * 182) + 1 ;
hence not 3 divides 547 by NAT_4:9; :: thesis:
547 = (5 * 109) + 2 ;
hence not 5 divides 547 by NAT_4:9; :: thesis:
547 = (7 * 78) + 1 ;
hence not 7 divides 547 by NAT_4:9; :: thesis:
547 = (11 * 49) + 8 ;
hence not 11 divides 547 by NAT_4:9; :: thesis:
547 = (13 * 42) + 1 ;
hence not 13 divides 547 by NAT_4:9; :: thesis:
547 = (17 * 32) + 3 ;
hence not 17 divides 547 by NAT_4:9; :: thesis:
547 = (19 * 28) + 15 ;
hence not 19 divides 547 by NAT_4:9; :: thesis:
547 = (23 * 23) + 18 ;
hence not 23 divides 547 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 547 & n is prime holds
not n divides 547 by XPRIMET1:18;
hence 547 is prime by NAT_4:14; :: thesis: