How did the Swiss family Robinson build their second home?

After their initial home, the treehouse, is damaged, the Robinsons set about to build a dwelling in a cave. They go about it as follows:
They bring a cart full of supplies with which to dig and hew, such as spades, pickaxes, and hammers, and they head for a large piece of rock. They then set to work on carving through the rock. After ten days of hard labor, they work through the face of the mountain enough to detect an interior cavern that they can feel when they stick a long pole through the little bit of remaining wall.
They manage to break into this cavern but are not yet home free, for it is filled with poisonous gas. They finally get rid of the gas by firing dozens of signal rockets into the space. These purify the air.
The family is then able to occupy this new space, which is spacious, dry, and beautiful.