Merging two incoming broadband lines for faster Internet

This is known as Multi-Homing or Multi-WAN. Most router manufacturer firmwares don't support this, but 3rd party firmware (DD-WRT, pfsense) is capable of doing load-balancing on a Multi-WAN connection.

The catch is while you can create 20Mbps of bandwidth, you cannot achieve 20Mbps of speed on a single connection. You would be able to have two independent 10Mbps streams, however.

To actually merge two connections into a single connection where you can push the combined bandwidth as if it's coming from a single pipe requires bonding, which would either need to be provided by your ISP if all the connections are with the same ISP, or by a 3rd party if the connections are to different ISPs or your ISP won't do bonding for you. It looks like shanabus' answer has some links that can help you explore that idea.

You could try a service that runs "Broadband Bonding" such as Mushroom Networks.

This may be effectively possible through software (such as Octopus+) running on a PC connected to multiple internet connections, but that would happen after your router so your diagram wouldn't fit. You would have two separate routers connecting to your ISPs then run those connections into your PC.

Some great discussion on this thread talks about different setups that might get you going in the right direction.

Wingate supports multiple outgoing connections. You can set up multiple outgoing connections as either fallback, or bundle them to one big pipe:

Provide secure and managed Internet access for your entire network via a single or multiple shared internet connections

This is a software-only solution (apart from the extra Ethernet card you would need), price depends on your network size.