Bijective group homomorphism/Inverse mapping/Homomorphism/Fact/Proof

Proof

This follows from

$\begin{array}{cc}{\phi }^{-1}(h_1{h}_2)& ={\phi }^{-1}(\phi ({\phi }^{-1}(h_1))\phi ({\phi }^{-1}(h_2)))\\ & ={\phi }^{-1}\left(\phi ({\phi }^{-1}(h_1){\phi }^{-1}(h_2))\right)\\ & ={\phi }^{-1}(h_1){\phi }^{-1}(h_2).\end{array}$

To fact