now :: thesis: ( not 2 divides 137 & not 3 divides 137 & not 5 divides 137 & not 7 divides 137 & not 11 divides 137 )

137 = (2 * 68) + 1 ;
hence not 2 divides 137 by NAT_4:9; :: thesis: ( not 3 divides 137 & not 5 divides 137 & not 7 divides 137 & not 11 divides 137 )
137 = (3 * 45) + 2 ;
hence not 3 divides 137 by NAT_4:9; :: thesis: ( not 5 divides 137 & not 7 divides 137 & not 11 divides 137 )
137 = (5 * 27) + 2 ;
hence not 5 divides 137 by NAT_4:9; :: thesis: ( not 7 divides 137 & not 11 divides 137 )
137 = (7 * 19) + 4 ;
hence not 7 divides 137 by NAT_4:9; :: thesis: not 11 divides 137
137 = (11 * 12) + 5 ;
hence not 11 divides 137 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 137 & n is prime holds
not n divides 137 by XPRIMET1:10;
hence 137 is prime by NAT_4:14; :: thesis: verum