will formatting local disk C lead to lose of internet connection ?

[Papa Lazarous]

Re-installing windows on a reformatted disk will mean that you will have to set up your internet connection again but depending on the router you use, it will not necessarily be difficult.
If you are using a router that was supplied by your ISP as part of your service then these are pre-configured and as soon as you reconnect the router you will just have to follow a few on screen prompts to get up and running on the internet again. [like Connor's answer says]
If you have supplied your own router than you will need to insert the CD that it came with [you can also download the software required for set up from the manufacturers support site]. So if you don't have the CD and you purchased the router yourself, then make sure you download the setup software before you reformat and re-install windows whilst you have a working internet connection.
You will need to manually insert the username and password used for connecting to your ISP's service and wifi encryption settings, SSID etc.
You can access the setup and configuration for most routers by typing "http://192.168.0.1" into the address bar of your browser [or refer to the manual or online documentation, on the manufacturers support site for you make and model]