now :: thesis: ( not 2 divides 67 & not 3 divides 67 & not 5 divides 67 & not 7 divides 67 )

67 = (2 * 33) + 1 ;
hence not 2 divides 67 by NAT_4:9; :: thesis: ( not 3 divides 67 & not 5 divides 67 & not 7 divides 67 )
67 = (3 * 22) + 1 ;
hence not 3 divides 67 by NAT_4:9; :: thesis: ( not 5 divides 67 & not 7 divides 67 )
67 = (5 * 13) + 2 ;
hence not 5 divides 67 by NAT_4:9; :: thesis: not 7 divides 67
67 = (7 * 9) + 4 ;
hence not 7 divides 67 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 67 & n is prime holds
not n divides 67 by XPRIMET1:8;
hence 67 is prime by NAT_4:14; :: thesis: verum