Using embedding of ${L^\infty }(0,1)$ in ${H^1}(0,1)$

Question

I want to ask you about a proof that I found strange. Let $f,g \in H_0^2(0,L)$ and let $$h = {\left( {{f_x}} \right)^2} - {\left( {{g_x}} \right)^2}$$ we have $${\left\| h \right\|_{H_0^1(0,1)}} = {\left\| {{{\left( {{f_x}} \right)}^2} - {{\left( {{g_x}} \right)}^2}} \right\|_{H_0^1(0,1)}} = {\left\| {\left( {{f_x} - {g_x}} \right)\left( {{f_x} + {g_x}} \right)} \right\|_{H_0^1(0,1)}}$$ Can I say that $${\left\| {\left( {{f_x} - {g_x}} \right)\left( {{f_x} + {g_x}} \right)} \right\|_{H_0^1(0,1)}} \leqslant {\left\| {\left( {{f_x} + {g_x}} \right)} \right\|_{{L^\infty }(0,1)}}{\left\| {\left( {{f_x} - {g_x}} \right)} \right\|_{H_0^1(0,1)}}$$ ? thanks.