I am reading Munkres Topology and following given Urysohn metrization theorem and its proof, but can't understand why the injectivity simply given after define a index functions and product them.

I had highlighted the point that I am missing at the below inserted images. 

[![enter image description here](https://i.stack.imgur.com/YCa5o.png)](https://i.stack.imgur.com/YCa5o.png)[![enter image description here](https://i.stack.imgur.com/veZ0B.png)](https://i.stack.imgur.com/veZ0B.png)