now :: thesis: ( not 2 divides 173 & not 3 divides 173 & not 5 divides 173 & not 7 divides 173 & not 11 divides 173 & not 13 divides 173 )

173 = (2 * 86) + 1 ;
hence not 2 divides 173 by NAT_4:9; :: thesis: ( not 3 divides 173 & not 5 divides 173 & not 7 divides 173 & not 11 divides 173 & not 13 divides 173 )
173 = (3 * 57) + 2 ;
hence not 3 divides 173 by NAT_4:9; :: thesis: ( not 5 divides 173 & not 7 divides 173 & not 11 divides 173 & not 13 divides 173 )
173 = (5 * 34) + 3 ;
hence not 5 divides 173 by NAT_4:9; :: thesis: ( not 7 divides 173 & not 11 divides 173 & not 13 divides 173 )
173 = (7 * 24) + 5 ;
hence not 7 divides 173 by NAT_4:9; :: thesis: ( not 11 divides 173 & not 13 divides 173 )
173 = (11 * 15) + 8 ;
hence not 11 divides 173 by NAT_4:9; :: thesis: not 13 divides 173
173 = (13 * 13) + 4 ;
hence not 13 divides 173 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 173 & n is prime holds
not n divides 173 by XPRIMET1:12;
hence 173 is prime by NAT_4:14; :: thesis: verum