now :: thesis: ( not 2 divides 167 & not 3 divides 167 & not 5 divides 167 & not 7 divides 167 & not 11 divides 167 )

167 = (2 * 83) + 1 ;
hence not 2 divides 167 by NAT_4:9; :: thesis: ( not 3 divides 167 & not 5 divides 167 & not 7 divides 167 & not 11 divides 167 )
167 = (3 * 55) + 2 ;
hence not 3 divides 167 by NAT_4:9; :: thesis: ( not 5 divides 167 & not 7 divides 167 & not 11 divides 167 )
167 = (5 * 33) + 2 ;
hence not 5 divides 167 by NAT_4:9; :: thesis: ( not 7 divides 167 & not 11 divides 167 )
167 = (7 * 23) + 6 ;
hence not 7 divides 167 by NAT_4:9; :: thesis: not 11 divides 167
167 = (11 * 15) + 2 ;
hence not 11 divides 167 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 167 & n is prime holds
not n divides 167 by XPRIMET1:10;
hence 167 is prime by NAT_4:14; :: thesis: verum