967 = (2 * 483) + 1 ;
hence not 2 divides 967 by NAT_4:9; :: thesis:
967 = (3 * 322) + 1 ;
hence not 3 divides 967 by NAT_4:9; :: thesis:
967 = (5 * 193) + 2 ;
hence not 5 divides 967 by NAT_4:9; :: thesis:
967 = (7 * 138) + 1 ;
hence not 7 divides 967 by NAT_4:9; :: thesis:
967 = (11 * 87) + 10 ;
hence not 11 divides 967 by NAT_4:9; :: thesis:
967 = (13 * 74) + 5 ;
hence not 13 divides 967 by NAT_4:9; :: thesis:
967 = (17 * 56) + 15 ;
hence not 17 divides 967 by NAT_4:9; :: thesis:
967 = (19 * 50) + 17 ;
hence not 19 divides 967 by NAT_4:9; :: thesis:
967 = (23 * 42) + 1 ;
hence not 23 divides 967 by NAT_4:9; :: thesis:
967 = (29 * 33) + 10 ;
hence not 29 divides 967 by NAT_4:9; :: thesis:
967 = (31 * 31) + 6 ;
hence not 31 divides 967 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 967 & n is prime holds
not n divides 967 by XPRIMET1:22;
hence 967 is prime by NAT_4:14; :: thesis: